# Long-Tailed Recognition via Weight Balancing

Shaden Alshammari$\ast$ Yu-Xiong Wang$\natural$ Deva
Ramanan$\sharp$,${\dagger}$ Shu Kong$\sharp$
$\ast$MIT $\natural$UIUC ${\dagger}$Argo AI $\sharp$CMU
<EMAIL_ADDRESS><EMAIL_ADDRESS>{deva<EMAIL_ADDRESS>
https://github.com/ShadeAlsha/LTR-weight-balancing

###### Abstract

In the real open world, data tends to follow long-tailed class distributions,
motivating the well-studied long-tailed recognition (LTR) problem. Naive
training produces models that are biased toward common classes in terms of
higher accuracy. The key to addressing LTR is to balance various aspects
including data distribution, training losses, and gradients in learning. We
explore an orthogonal direction, weight balancing, motivated by the empirical
observation that the naively trained classifier has “artificially” larger
weights in norm for common classes (because there exists abundant data to
train them, unlike the rare classes). We investigate three techniques to
balance weights, L2-normalization, weight decay, and MaxNorm. We first point
out that L2-normalization “perfectly” balances per-class weights to be unit
norm, but such a hard constraint might prevent classes from learning better
classifiers. In contrast, weight decay penalizes larger weights more heavily
and so learns small balanced weights; the MaxNorm constraint encourages
growing small weights within a norm ball but caps all the weights by the
radius. Our extensive study shows that both help learn balanced weights and
greatly improve the LTR accuracy. Surprisingly, weight decay, although
underexplored in LTR, significantly improves over prior work. Therefore, we
adopt a two-stage training paradigm and propose a simple approach to LTR: (1)
learning features using the cross-entropy loss by tuning weight decay, and (2)
learning classifiers using class-balanced loss by tuning weight decay and
MaxNorm. Our approach achieves the state-of-the-art accuracy on five standard
benchmarks, serving as a future baseline for long-tailed recognition.

(a) per-class classification accuracy vs. class cardinality on CIFAR100-LT
(imalance factor 100)

(b) norms of per-class weights from the learned classifier vs. class
cardinality

Figure 1: Long-tailed recognition (LTR) requires training on long-tailed class
distributed data (black curve in (a)). (a) Networks naively trained on such
data are biased toward common classes in terms of higher accuracy (orange
bars). (b)-left plots the L2 norms of per-class weights in the naive
classifier. Clearly, classifiers of common classes have “artificially” larger
norms than rare ones, because they are trained with more data. This can result
in over-predictions of common classes, or alternatively, under-predictions of
rare classes. This observation motivates us to balance norms via parameter
regularization. To do so, we explore simple weight balancing techniques
including L2-normalization, weight decay, and the MaxNorm constraint. We find
that applying the latter two results in class weights to be far more balanced
((b)-right), allowing rare classes to have a “fighting chance” when competing
with common classes. Our model boosts overall accuracy to 53.35% (blue bars in
(a)), significantly higher than the naive model (38.38%) and prior art, e.g.,
RIDE (49.1%) [73], ACE (49.6%) [10], and PaCo (52.0%) [17]. Results are from
experiments (Table 1) on CIFAR100-LT with an imbalance factor 100 [12].

##  1 Introduction

In the real open world, data tends to follow long-tailed distributions [60, 8,
85, 84]. Through the lens of classification, this means that the number of
per-class data, or class cardinality, is heavily imbalanced [27, 72]. Numerous
applications emphasize the rare classes. For example, autonomous vehicles
should recognize not only common objects such as cars and pedestrians, but
also rare ones like strollers and animals for driving safety [41]. A bio-image
analysis system should recognize both commonly- and rarely-seen species for
ecological research [63, 72]. This motivates the well-studied problem of long-
tailed recognition (LTR), which trains on class-imbalanced data and aims to
achieve high accuracy averaged across all the classes [84]. LTR has attracted
increasing attention especially using deep neural networks [12, 38, 78].

Status quo. Because common classes have significantly more training data than
rare classes, they dominate the training loss, contribute the most of
gradients, and obtain high accuracy [84]. Consequently, a naively trained
model performs well on them but significantly worse on the rare classes (Fig.
1a). The key to addressing LTR is to balance various aspects. Many methods
propose to balance per-class data distributions during training by upsampling
rare classes or downsampling common classes [14, 22, 23]. Some others balance
the losses or gradients during training [71, 40, 19, 12]. Some approaches
adopt transfer learning that learn features on common classes and use the
features to learn rare-class classifiers [74, 87, 49, 37]. It shows that
decoupling feature learning and classifier learning leads to significant
improvement over models that train them jointly [38]. From benchmarking
results, the state-of-the-art accuracy is achieved by either ensembling expert
models [73, 76, 10, 26, 23] or the adoption of self-supervised pretraining
with aggressive data augmentation techniques [17].

Motivation. We observe that a naively trained model on long-tailed class
distributed data has “artificially” large weights for common classes (Fig 1b).
Prior work also notes this observation [38]. Intuitively, this is because
common classes have more training data that significantly grows classifier
weights (Fig. 2a). This motivates our work to balance network weights across
classes for long-tailed recognition. In contrast to existing methods (as
exhaustively reviewed in a recent survey paper [84]), our work explores an
orthogonal direction of _weight balancing_.

Contribution. To balance network weights in norm, we study three simple
techniques. We first point out that L2-normalization perfectly balances
classifier weights to have unit norm (Fig. 2b). However, L2-normalization
might be too strict to learn flexible parameters for better classifiers. We
then study _weight decay_ [29, 44] and the _MaxNorm_ constraint [66, 35].
Weight decay penalizes larger weights more heavily and so learns small
balanced weights (Fig. 2c); MaxNorm encourages growing small weights within a
norm ball and caps all the weights by the radius (Fig. 2d). We find that both
effectively learn balanced weights and boost LTR performance, although these
well-known regularizers are _underexplored_ in the LTR literature. Please
refer to Fig. 1 for a nutshell of our work.

Key Findings. We show how simple regularizers boost LTR performance. Without
inventing new losses or adopting aggressive augmentation techniques or
designing new network modules, we follow the simple two-stage training
paradigm [38] and derive a simple approach that rivals or outperforms the
state-of-the-art methods: (1) train a backbone using the standard cross-
entropy loss by properly tuning _weight decay_ , and (2) train the classifier
using a class-balanced loss by tuning _weight decay_ and _MaxNorm_. It is
important to note how our simple approach challenges the increasingly
complicated LTR models, and hence serves as a strong future baseline for LTR.

##  2 Related Work

Long-Tailed Recognition (LTR). Real-world data tends to follow long-tailed
class distributions, i.e., a few classes are commonly seen that have
significantly more data than many classes that are infrequently / rarely seen.
As a result, a model naively trained on such data performs significantly worse
on rare classes than common classes. LTR requires training on such data to
achieve high accuracy averaged across all classes [12, 38, 78]. For LTR,
numerous methods emphasize the accuracy on rare-classes. Data re-balancing
techniques resample the training data to achieve a more balanced data
distribution across classes [67, 53], such as over-sampling rare-classes [14,
28] and undersampling common-classes [21]. Class-balanced loss reweighting
assigns weights to the classes [19, 40, 12, 39, 36, 83], or even training
examples [47, 68, 62, 39], aiming to modify their gradients to make the class-
imbalanced data contribute properly to training. Transfer learning methods
transfer feature representations learned on the common-classes to the rare-
classes [79, 48]. Recent work examines the training procedure and finds LTR to
be better addressed by decoupling feature learning and classifier learning,
rather than training them jointly [38, 88]. It is found that the SGD momentum
causes issues in LTR that prevent further improvement [71] . Other
sophisticated methods exploit self-supervised pretraining with more aggressive
data augmentation techniques [17], or ensemble expert models trained on
different data regimes [10, 73]. For a comprehensive review of the LTR
literature, we refer the reader to the recent survey paper [84]. Different
from all the existing methods, we explore an orthogonal direction of parameter
regularization, leading to a much simpler approach to LTR.

How are per-class weight norms evolving in training ($x$-axis)? Classes are
sorted w.r.t. cardinality ($y$-axis).
(a) naive (b) L2-normalization (c) WD (d) MaxNorm (e) WD+MaxNorm

Figure 2: (a) In a naive model, all weights grow in norm during training,
while those of common classes grow much faster. (b) Because L2-normalization
constrains weights to be unit-norm, weight norms stay constant during
training. (c) Weight decay (WD) regularizes all weights to be small while
still allowing them to grow. (d) MaxNorm caps large weights (of common
classes) while letting small weights grow. (e) Combining weight decay and
MaxNorm results in small and balanced weights in norm. All plots share the
same color map.

Parameter Regularization adds extra information to solve an ill-posed problem,
improving generalizability and preventing overfitting [57, 7, 9].
Regularization plays a crucial role in deep learning [43]. One well-known
regularization is weight decay, which often applies L2-norm penalty on network
weights [29, 44, 52]. There exist many more regularizations [24, 45], such as
weight normalization [64, 1], MaxNorm constraints [66, 35, 25], data
augmentation [82], and dropout [35]. In this work, we particularly examine the
well-known yet underexplored regularizers in the LTR literature:
L2-normalization, weight decay, and the MaxNorm constraint [66, 35, 25].

Stage-wise Training turns to be effective in training deep networks [31, 50,
80, 86]. This can date back to stage-wise layer pretraining [34, 4]. Recently,
Kang et al. convincingly demonstrate that stage-wise training is important to
LTR [38]. Concretely, Kang et al. propose to decouple feature learning and
classifier learning into two independent stages [38]: (1) feature learning
using the standard cross-entropy loss, and (2) classifier learning over the
learned feature using a class-balancing loss. While they did not explain why a
single one-stage training with the class-balancing loss performed poorly,
intuitively, this is because a class-balancing loss artificially scales up
gradients computed from rare-class training data, which hurts the feature
representation learning and hence the final LTR performance. Follow-up work
indirectly demonstrates this intuition with improved performance by
stabilizing gradients during training [61, 71]. In our paper, we adopt this
two-stage training procedure, but focus on how to balance network weights for
LTR.

##  3 Weight Balancing for Long-Tailed Learning

Preliminaries. Long-tailed recognition (LTR) aims to train over a training set
${\cal D}$$=$$\\{({\bf x}_{i},y_{i})\\}_{i=1}^{N}$, where data example ${\bf
x}_{i}$ is labeled as $y_{i}$ $\in[1,\dots,K]$. For class-$k$, ${\cal D}_{k}$
is the set of all its training examples and $|{\cal D}_{k}|$ is its
cardinality. The imbalance factor, IF$=$$\frac{\max_{k}|{\cal
D}_{k}|}{\min_{k}|{\cal D}_{k}|}$, measures how imbalanced the long-tailed
training set is. For LTR, IF$\gg$$1$. LTR emphasizes classification accuracy
averaged over classes, i.e., accuracy=$\frac{1}{K}\sum_{k}$acck, where acck is
the accuracy computed over testing examples of class-$k$.

LTR focuses on learning a $K$-way classification network
$f(\cdot;\mbox{\boldmath$\Theta$\unboldmath})$ parameterized by
$\mbox{\boldmath$\Theta$\unboldmath}=\\{\mbox{\boldmath$\theta$\unboldmath}_{l,j}\\}$,
where $\mbox{\boldmath$\theta$\unboldmath}_{l,j}$ is the $j^{th}$ filter
weights at layer-$l$. In a conv-layer,
$\mbox{\boldmath$\theta$\unboldmath}_{l,j}$ is a 3D kernel that convolves the
input (activation). For brevity, we denote
$\mbox{\boldmath$\theta$\unboldmath}_{k}$ as the classifier filter
corresponding to class-$k$. Given a data example ${\bf x}_{i}$, the network
predicts a label $y_{i}^{\prime}=f({\bf
x}_{i};\mbox{\boldmath$\Theta$\unboldmath})$. We measure the prediction error
between $y^{\prime}_{i}$ and the ground-truth $y_{i}$ using a cost function
$\ell(y^{\prime}_{i},y_{i})$, e.g., a cross-entropy (CE) loss [7, 56] or a
class-balanced loss (CB) [19]. To train the network
$f(\cdot;\mbox{\boldmath$\Theta$\unboldmath})$, we optimize $\Theta$ by
minimizing $\ell(y^{\prime}_{i},y_{i})$ over the whole training set $\cal D$:

$\displaystyle{\mbox{\boldmath$\Theta$\unboldmath}^{*}}=\arg\min_{\mbox{\boldmath$\Theta$\unboldmath}}F(\mbox{\boldmath$\Theta$\unboldmath};{\cal
D})\equiv\sum_{i=1}^{N}\ell\big{(}f({\bf
x}_{i};\mbox{\boldmath$\Theta$\unboldmath}),y_{i}\big{)}.$ (1)

Naively solving (1) produces a classifier (i.e., the last layer) that has
artificially large weights in norm for common classes (Fig. 1b-left, Fig. 2a).
Therefore, we are motivated to learn a balanced classifier by regularizing
classifier weights, denoted by $\mbox{\boldmath$\theta$\unboldmath}_{k}$ for
$k=1,\dots,K$. Intermediate layers also have imbalanced filter weights (Fig.
3) even though a filter tends to fire on multiple classes [81, 2]. Generally,
one can also balance the weights at intermediate layers, and our study shows
that doing so boosts performance. Nevertheless, to simplify presentation in
the following, we focus on regularization on the classifier weights
$\mbox{\boldmath$\theta$\unboldmath}_{k}$’s.

###  3.1 Weight Balancing Techniques

We examine the following three techniques to balance weights with respect to
norms.

L2-normalization. A “perfect” way to balance the classifier weights
$\mbox{\boldmath$\theta$\unboldmath}_{k}$’s is to L2-normalize the classifier
weights:

$\displaystyle{\mbox{\boldmath$\Theta$\unboldmath}^{*}}=\arg\min_{\mbox{\boldmath$\Theta$\unboldmath}}F(\mbox{\boldmath$\Theta$\unboldmath};{\cal
D}),\quad\text{\emph{s.t.} \ \
}\|\mbox{\boldmath$\theta$\unboldmath}_{k}\|_{2}^{2}=1,\ \ \forall\ k.$ (2)

As L2-normalization forces weights to be unit-length, the classifier weights
will have unit norm constant during training (Fig. 2b). Inspired by [38], we
also post-hoc L2-normalize a trained classifier, i.e.,
$\mbox{\boldmath$\theta$\unboldmath}_{k}^{\prime}$$=$$\mbox{\boldmath$\theta$\unboldmath}_{k}/\|\mbox{\boldmath$\theta$\unboldmath}_{k}\|_{2}$.
We find that post-hoc L2-normalization oftentimes improves LTR performance,
favoring rare-classes yet sacrificing common-class accuracy. But it can also
significantly decrease overall performance, e.g., on iNaturalist in Table 3.
Post-hoc L2-normalization is similar to the $\tau$-normalization [38], which
allows varied per-class weight norms (rather than forcing them to be the same)
and achieves better LTR performance. This suggests that L2-normalization is
too strict to strike a balance among the long-tailed distributed classes.
Importantly, our exploration finds that, while training with an
L2-normalization constraint on the classifier improves over naive training, it
underperforms the other two regularizers described below.

Weight Decay is a well-studied technique [55, 44] used to constrain a network
by limiting the growth of the network weights. It decreases the complexity of
the network, effectively mitigating overfitting and improving generalization.
Weight decay typically applies an L2-norm penalty to the network weights (we
focus on the classifier $\mbox{\boldmath$\theta$\unboldmath}_{k}$’s for now):

$\displaystyle{\mbox{\boldmath$\Theta$\unboldmath}^{*}}=\arg\min_{\mbox{\boldmath$\Theta$\unboldmath}}F(\mbox{\boldmath$\Theta$\unboldmath};{\cal
D})+\lambda\sum_{k}\|\mbox{\boldmath$\theta$\unboldmath}_{k}\|_{2}^{2},$ (3)

where $\lambda$ is a hyperparameter to control the impact of weight decay. The
weight decay term in (3) penalizes more heavily on large weights, preventing
them from growing too large (Fig. 2c) [55, 44]. That said, weight decay
encourages learning small balanced weights, as demonstrated by Fig. 2.
Somewhat surprisingly, weight decay is underexplored in the literature of
long-tailed recognition. To the best of our knowledge, existing methods did
not properly tune weight decay [19, 71] (cf. code [11, 18, 70]) aside from
their technical innovations. This makes it unclear whether their improved LTR
performance is due to better regularization inherent in these methods.
Importantly, our exploration demonstrates that, by simply tuning weight decay,
we outperform most of the state-of-the-art methods on long-tailed benchmarks
(Tables 2 and 3)!

MaxNorm Constraint. The third regularizer we explore is the MaxNorm constraint
[66, 35, 25]. MaxNorm caps weight norms within an L2-norm ball with radius
$\delta$:

$\displaystyle{\mbox{\boldmath$\Theta$\unboldmath}^{*}}=\arg\min_{\mbox{\boldmath$\Theta$\unboldmath}}F(\mbox{\boldmath$\Theta$\unboldmath};{\cal
D}),\quad\text{\emph{s.t.} \ \
}\|\mbox{\boldmath$\theta$\unboldmath}_{k}\|_{2}^{2}\leq\delta^{2},\ \
\forall\ k,$ (4)

where the hyperparameter $\delta$ is the radius of the norm-ball. Solving (4)
can be efficiently done through Projected Gradient Descent (PGD), which
projects big weights that are outside the L2-norm ball onto the constraint set
[66]. It simply applies a renormalization step after each batch update.
Specifically, at each iteration, PGD first computes an updated
$\mbox{\boldmath$\theta$\unboldmath}_{k}$ and then projects it onto the norm
ball:

$\displaystyle\mbox{\boldmath$\theta$\unboldmath}_{k}\leftarrow\min\Big{(}1,{\delta}/{\|\mbox{\boldmath$\theta$\unboldmath}_{k}\|_{2}}\Big{)}*\mbox{\boldmath$\theta$\unboldmath}_{k}.$
(5)

Different from L2-normalization that strictly sets the norm value for all the
filter weights as 1, MaxNorm relaxes this constraint that allows the weights
to move within the norm-ball during training, as visualized in Fig. 2d.

###  3.2 Further Discussion

To better understand how and why the aforementioned regularizers work for
long-tailed recognition, we discuss the following aspects.

Figure 3: Weight decay helps learn balanced weights at hidden layers. We
compare the norm distribution at each layer (which has 512 filters) from the
naive model (orange) and the one trained with weight decay (blue). For each
layer of a model, we sort the filter weights of each layer from high to low,
compute their mean (the centerline) and variance (the shadow). While
individual filters in the hidden layers are not class-specific by design,
recent work demonstrates that certain filters tend to fire on certain classes
[81, 2]; we still find them to be “imbalanced” in norms from the naive model.
Weight decay encourages learning small and balanced filters, cf. its flat
centerline and small variance.

Weight Decay and MaxNorm. Both regularizers balance weight norms _dynamically_
during training, as opposed to L2-normalization which simply forces per-filter
weights to be unit-length in norm. Weight decay encourages learning small
weights, and MaxNorm encourages weights to grow within a norm ball but cap
them when their norms exceed the radius. Weight decay pulls all weights to the
origin. As a result, when $\lambda$ increases in (3), the weight decay penalty
prevails $F(\mbox{\boldmath$\Theta$\unboldmath};{\cal D})$, making training
unstable [6] (Fig. 4). In contrast, MaxNorm does not pull weights towards the
origin but simply caps the weight norms, and so has better numerical
stability.

Although weight decay and MaxNorm appear to be quite different, they are
related that weight decay can be thought of as an immediate step when solving
MaxNorm. Let’s rewrite the MaxNorm constrained objective function (4) by
constructing a Lagrangian function:

$\displaystyle\mbox{\boldmath$\Theta$\unboldmath}^{*}=\arg\min_{\mbox{\boldmath$\Theta$\unboldmath}}\max_{\gamma\geq
0}F(\mbox{\boldmath$\Theta$\unboldmath};{\cal
D})+\sum_{k}\gamma(\|\mbox{\boldmath$\theta$\unboldmath}_{k}\|_{2}^{2}-\delta),$
(6)

where $\gamma$ is the Karush–Kuhn–Tucker (KKT) multiplier. Suppose that we
could solve (6) using the coordinate descent method, i.e., iteratively
optimizing over $\Theta$ and $\gamma$ [75]. When fixing $\gamma$, we have the
same loss as (3) which is constrained by weight decay, and $\gamma$ becomes
the hyperparameter $\lambda$ to control weight decay. That said, solving the
weight decay constrained problem (3) is a step of solving MaxNorm (4).
Interestingly, we find that applying weight decay and MaxNorm jointly yields
better performance than using each of them independently. This is probably
because of their complementary advantages: (1) weight decay on the small
weights still improves their generalization and reduces overfitting, and (2)
MaxNorm prevents the large weights from dominating the training.

Figure 4: Tuning weight decay drastically improves long-tailed recognition
performance. We do not use any class-balancing techniques but simply use CE
loss and tune weight decay $\lambda$ to regularize all network weight. For
example, tuning $\lambda$ yields $46.1\%$ accuracy on CIFAR100-LT (IF=100),
outperforming many state-of-the-art methods such as DiVE (45.4%) [32] and SSD
(46.0%) [46]. By checking the publicly available code, we find that existing
methods do not tune weight decay, e.g., [19, 12] set $\lambda=$2e-4 (according
to their code [11, 18]), leading to poor accuracy 38.32%.

Extreme cases. When $\delta$$\rightarrow$$\infty$ in MaxNorm, (4) boils down
to the naive training (1). On the other hand, a sufficiently small $\delta$
encourages all the weights to be close to the surface of the norm-ball. This
is still different from the L2-normalization which strictly requires the
weights to be on the surface. Compared to L2-normalization (Fig. 2b), MaxNorm
offers freespace within the norm ball to let weights grow (Fig. 2d). This
intuitively explains why MaxNorm performs better than L2-normalization.

Weight decay can easily balance all network weights. We point out that weight
decay regularizes classifier weights without the need to separate per-class
filters. This offers convenience in training, differently from MaxNorm which
must separate each filter and scaling it w.r.t its norms. Because of such a
convenience, weight decay can be easily used to balance all network weights
(Fig. 3). In principle, MaxNorm can also be applied to all layers, but we find
it non-trivial to do so, as this seems to require setting per-layer thresholds
in (4) (tuning which is time-consuming). While weight decay is widely used in
network training, we find that properly tuning it drastically improves long-
tailed recognition accuracy (Table 1).

###  3.3 Training Pipeline

Because the aforementioned weight balancing techniques are not exclusive to
each other, in principle, one can use a single technique or multiple ones
together. Recall that we follow the two-stage training paradigm [38] in our
work, which first trains a network for feature representation and then trains
the classifier atop the learned features. This raises a question how to apply
the weight balancing techniques effectively. Among extensive exploration, we
find that tuning $\lambda$ for weight decay in (3) is sufficient to learn a
generalizable feature representation as the first-stage training. In contrast,
applying MaxNorm is nontrivial because we find that it requires setting per-
layer thresholds in (4). This tuning process is time-consuming. In the second-
stage training (i.e., training the classifier), we find that tuning
either/both weight decay and MaxNorm remarkably improves LTR accuracy. Because
the classifier training simply involves only one layer (or two layers if we
think of the top two as a non-linear classifier), tuning hyperparameters of
the regularizers is quite efficient. To tune them, one can use random search
[5] or Bayesian Optimization [69, 58]. We use the latter in this work. In
summary, our simple training pipeline consists of the following two stages:

  1. 1.

_Feature learning_ : train a network by using the cross-entropy loss and
tuning weight decay.

  2. 2.

_Classifier learning_ : train a classifier over the learned features using a
class-balanced loss [19], weight decay, and MaxNorm.

Figure 5: Frequency distributions w.r.t class cardinality of five benchmarks.
Left: We modify CIFAR100 by downsampling examples per class with different
imbalance factors (IF) varying from 10 to 100. Right: We use two large-scale
datasets: ImageNet-LT [49] that downsamples per-class images from ImageNet
[20], and iNaturalist [72] which is a real-world dataset with IF=500.

##  4 Experiments

We carry out extensive experiments to demonstrate how balancing network
weights boosts long-tailed recognition performance. First, we ablate the
design choices in our pipeline as suggested in Section 3.3. Then, we benchmark
our methods on five established long-tailed datasets, showing that they rival
or outperform existing LTR methods. We start with the experiment setup.

Figure 6: Per-class weight norms (top row) and marginal likelihood (bottom
row) in the classifier vs. class ID sorted by class cardinality in decreasing
order. The plots are on CIFAR100-LT (IF100) val-set which has class-balanced
data. According to [61], the ideal marginal likelihood should follow a uniform
distribution. Interestingly, L2-normalization that “perfectly” balances weight
norms does not produce “uniform” marginal likelihood. Weight decay (WD)
slightly mitigates norm imbalance and marginal likelihood imbalance, but
MaxNorm dramatically helps both. The final model that incorporates MaxNorm,
weight decay, and class-balanced loss yields nearly “perfect” marginal
likelihood and balanced weights, which have a small bias towards rare-classes,
presumably to emphasize their accuracy.

###  4.1 Experiment Setup

Datasets. We use five long-tailed benchmarks. Following [13], we modify the
CIFAR100 dataset [42] by downsampling per-class training examples using some
exponential decay functions, resulting in a long-tailed version, named
_CIFAR100-LT_. CIFAR100-LT still has 100 classes and a balanced validation set
for evaluation. By varying an imbalance factor (IF) $\in[100,50,10]$, we
create three long-tailed training sets (Fig. 5-left). _ImageNet-LT_ is
introduced in [48] by artificially truncating the balanced version ImageNet
[20]. ImageNet-LT has 1,000 classes, and the number of per-class training data
ranges from 5 to 1280. _iNaturalist2018_ [72] is a real-world dataset that has
8,142 naturally long-tailed classes. Fig. 5 summarizes the class frequency
distributions of these datasets. ImageNet and iNaturalist2018 are publicly
available for non-commercial research and educational purposes; CIFAR100 is
released under the MIT license. We note that ImageNet and CIFAR100 have a
“people” class or contain images that captured human faces and person
signatures. This is a concern related to fairness and privacy. Therefore, we
cautiously proceed our research and release our code under the MIT License
without re-distributing the data.

Network architectures. For a fair comparison to prior art, we follow [19, 38,
49, 37, 78] to use specific network architectures on each dataset. We use
ResNet32 [31] on CIFAR100-LT, ResNeXt50 [77] on ImageNet-LT, and ResNet50 [31]
on iNaturalist2018.

Evaluation protocol. On each dataset, we train on the long-tailed class-
imbalanced training set and evaluate on its (balanced) validation/test set. On
ImageNet-LT, we tune hyperparameters and select models on its val-set and
report performance on the test-set. On CIFAR100-LT and iNaturalist, which only
have train-val sets, we follow the literature [49] that uses the val-sets to
tune and benchmark. Following [49], we further report accuracy on three splits
of classes that have varied numbers of training data: _Many_ ($>$100),
_Medium_ (20$\sim$100), and _Few_ ($<20$).

Implementation. We train our models using PyTorch toolbox [59] on GeForce GTX
2080Ti GPUs. The total time spent on this work is $\sim$2 GPU years with
respect to this GPU type. We train each model for 200 epochs, with batch size
as 64 (for CIFAR and ImageNet-LT) / 512 (for iNaturalist), SGD optimizer with
momentum 0.9, and cosine learning rate scheduler [51] that gradually decays
learning rates from 0.01 to 0. We also use random left-right flipping and
cropping as our training augmentation.

###  4.2 Ablation Study

We study (1) the impact of weight decay in LTR, (2) how to regularize
classifier learning, (3) classifier weight norms and marginal likelihood
distribution, and (4) the evolution of weight norms during training. We use
CIFAR100-LT (IF=100) for this study (unless stated otherwise).

Weight decay. We set a single constant $\lambda$ for all network parameters
and focus on the first-stage training only, i.e., we use the standard cross-
entropy loss to train a single network for classification. Fig. 4 draws the
top-1 accuracy as a function of $\lambda$ on the validation sets of three
benchmarks. Clearly, tuning $\lambda$ boosts accuracy, even outperforming many
state-of-the-art methods (cf. Tables 2 and 3)! Moreover, the optimal $\lambda$
varies for different datasets – larger datasets need a smaller weight decay,
intuitively because learning over more data helps generalization and so needs
less regularization.

How to regularize classifier learning. To study how to apply the balancing
techniques in the second-stage learning for classifiers, we also include
$\tau$-normalization [38] because it is an effective non-learned technique
that post-hoc scales the classifier learned in the first stage. We present
salient conclusions based on the results in Fig. 6 (more in the supplement).
First, with an improved backbone (owing to a properly tuned weight decay in
the first stage), $\tau$-norm boosts from 42.00% to 51.31%! This demonstrates
the importance of learning a backbone that has balanced weights (Fig. 3).
Second, it is crucial to use a class-balanced (CB) loss [19] to learn the
classifier. However, solely using the CB loss without regularizers only
slightly improves (from 46.08% to 47.09%); once regularized with weight decay,
it boosts to 52.42%. Third, applying both MaxNorm and weight decay improves
further (53.35%), and learning more layers (as a non-linear MLP classifier)
improves to 53.55%.

Classifier’s weight norms and marginal likelihood. Inspired by [61], we
examine the marginal likelihood based on predictions on the (balanced) test-
set, on which the ideal marginal likelihood follows a uniform distribution
[61]. We plot the marginal likelihood in Fig. 6, alongside the norm
distribution of different models. Interestingly, L2-normalization that
“perfectly” balances classifier weights does not produce balanced marginal
likelihood. In contrast, MaxNorm significantly helps learn balanced weights
and balanced marginal likelihood. Combining MaxNorm, weight decay, and the CB
loss, the model makes nearly “perfect” marginal likelihood with a small bias
toward rare-classes in weight norms, presumably because it learns to emphasize
rare-class accuracy.

Weight norm evolution during training. Fig. 2 depicts how classifier’s weight
norms evolve during training for different models. Briefly, without
regularization, weights in the naive model grow fast in norm. In contrast,
weight decay prevents weights from growing too large, and MaxNorm quickly caps
weights on a norm-ball surface and allows small weights to grow within the
ball.

Table 1: Ablation study on CIFAR100-LT (IF=100) w.r.t top-1 accuracy (%). “CE”: cross-entropy loss; “CB”: class-balanced loss [19]; “WD”: weight decay; “Max”: MaxNorm constraint; “$\tau$-norm”: $\tau$-normalization [38]; “+”: fine-tuning the last layer(s) as the second-stage training. Here are salient conclusions. (1) Learning with a properly tuned WD boosts performance from 38.38% to 46.08%, which is +8% increase. (2) Re-training the last layer with CB and WD gives another boost (+6%) to 52.42%. (3) Based on the above, applying additional MaxNorm yields a slight improvement +1% (53.35%); finetuning the last two layers achieves 53.55%. (4) Finetuning more layers performs worse (cf. the supplement), presumably because CB induces modified gradients that affect feature learning and so hurt the final LTR performance.  Model | Many | Medium | Few | All
---|---|---|---|---
on the last layer (classifier)
WD=0 (w/ CE) | 64.05 | 35.80 | 11.43 | 38.38
\+ $\tau$-norm | 59.54 | 38.23 | 25.93 | 42.00
WD tuned (w/ CE) | 76.94 | 44.28 | 12.17 | 46.08
\+ $\tau$-norm | 73.11 | 47.69 | 30.10 | 51.31
\+ L2norm | 76.09 | 47.74 | 20.87 | 49.60
\+ CE & L2norm | 76.37 | 48.11 | 21.00 | 49.87
\+ CE & WD | 76.97 | 45.94 | 14.00 | 47.22
\+ CB | 77.00 | 45.89 | 13.60 | 47.09
\+ CB & L2norm | 76.43 | 48.20 | 21.60 | 50.10
\+ CB & WD | 72.77 | 49.74 | 31.80 | 52.42
\+ CB & Max | 76.49 | 49.23 | 20.67 | 50.20
\+ CB & WD & Max | 72.60 | 51.86 | 32.63 | 53.35
on the last two layers
\+ CB & WD & Max | 71.37 | 51.17 | 35.53 | 53.55
Table 2: Benchmarking on CIFAR-100-LT with different imbalance factors $[100,50,10]$ w.r.t top-1 accuracy (%). Please refer to the caption of Table 1 for abbreviations; “+” methods use CB loss. WD makes a substantial impact on the training of LTR networks. Finetuning the classifier with proper regularization improves much further. This clearly shows the significance of parameter regularization in balancing weights for LTR. Somewhat surprisingly, properly tuning weight decay in the two-stage training paradigm outperforms all existing methods on these three datasets.  imbalance factor | 100 | 50 | 10
---|---|---|---
CE [19] | 38.32 | 43.85 | 55.71
CE+CB [19] | 39.60 | 45.32 | 57.99
KD [33] | 40.36 | 45.49 | 59.22
LDAM-DRW [12] | 42.04 | 46.62 | 58.71
BBN [88] | 42.56 | 47.02 | 59.12
LogitAjust [54] | 42.01 | 47.03 | 57.74
LDAM+SSP [78] | 43.43 | 47.11 | 58.91
Focal [47] | 38.41 | 44.32 | 55.78
Focal+CB [19] | 39.60 | 45.17 | 57.99
De-confound [71] | 44.10 | 50.30 | 59.60
$\tau$-norm [38] | 47.73 | 52.53 | 63.80
SSD [46] | 46.00 | 50.50 | 62.30
DiVE [32] | 45.35 | 51.13 | 62.00
DRO-LT [65] | 47.31 | 57.57 | 63.41
PaCo [17] | 52.00 | 56.00 | 64.20
ACE (4-expert) [10] | 49.60 | 51.90 | —
RIDE (4-expert) [73] | 49.10 | — | —
Our methods (weight balancing)
naive | 38.38 | 43.99 | 57.31
WD | 46.08 | 52.71 | 66.03
\+ L2norm | 49.60 | 56.33 | 67.16
\+ $\tau$-norm | 51.31 | 57.65 | 67.79
\+ WD | 52.42 | 57.47 | 67.96
\+ Max | 50.24 | 56.06 | 67.10
\+ WD & Max | 53.35 | 57.71 | 68.67

###  4.3 Benchmark Results

Compared Methods. Considering the rapid evolution of the LTR field [84], we
compare against most relevant methods. We choose methods that are recently
published and representative of different types, such as Focal [47] for loss
reweighting, PaCo [17] for self-supervised pretraining and aggressive data
augmentation, RIDE [73] for ensembling expert models, SSD [46] and DiVE [32]
for transfer learning, etc. For comparison, we report our methods including
the naive model, the one trained with properly tuned weight decay, and models
that have the second-stage learning for classifier with regularizers. Tables 2
and 3 list benchmarking results on the CIFAR100-LT datasets, and ImageNet-LT
and iNaturalist, respectively.

Results. Without bells and whistles, simply tuning weight decay (WD) in the
first-stage training significantly boosts LTR performance over naive training
and outperforms many prior methods. For example, on CIFAR100-LT (IF100) in
Table 2, our WD model achieves 46.08%, outperforming the naive model (38.38%)
and most of the compared methods including SSD (46.00%) [46] and DiVE (45.35%)
[32]. With the second stage (classifier learning), simply post-hoc modifying
(without learning) the classifier (learned in the first stage) significantly
improves performance from 46.09% to 49.60% (by L2-normalization) and to 51.31%
(by $\tau$-normalization). By learning the classifier regularized with MaxNorm
and/or weight decay, we achieve the state of the art (53.35%). Such a
conclusion holds on all benchmarks. However, on the two large-scale datasets
ImageNet-LT and iNaturalists in Table 3, our methods rival prior art but
underperform two types of methods that have “bells and whistles”, including
ensemble methods (RIDE [73] and ACE [46]) that learn and fuse multiple models,
and self-supervised learning based methods (PaCo [17] and SSD [46]) that adopt
aggressive data augmentation techniques [30, 16].

Table 3: Benchmarking on ImageNet-LT and iNaturalists in top-1 accuracy (%). Please refer to Table 2 for methods’ names and salient conclusions. We list the numbers of compared methods reported in their respective papers. Overall, our simple approach achieves competitive results with the prior methods particularly when they train a “single (expert)” model, although underperforms a few recent state-of-the-art methods which train and ensemble expert models (RIDE [73] and ACE [10]), or adopt self-supervised pretraining (e.g., PaCo [17] and SSD [46]) with aggressive data augmentation techniques [30, 16].  | ImageNet-LT |  | iNaturalist
---|---|---|---
| Many | Med. | Few | All |  | Many | Med. | Few | All
CE [38] | 65.9 | 37.5 | 7.7 | 44.4 |  | 72.2 | 63.0 | 57.2 | 61.7
CE+CB [19] | 39.6 | 32.7 | 16.8 | 33.2 |  | 53.4 | 54.8 | 53.2 | 54.0
KD [33] | 58.8 | 26.6 | 3.4 | 35.8 |  | 72.6 | 63.8 | 57.4 | 62.2
Focal [19] | 36.4 | 29.9 | 16.0 | 30.5 |  | — | — | — | 61.1
OLTR [49] | 43.2 | 35.1 | 18.5 | 35.6 |  | 59.0 | 64.1 | 64.9 | 63.9
LFME [76] | 47.1 | 35.0 | 17.5 | 37.2 |  | — | — | — | —
BBN [88] | — | — | — | — |  | 49.4 | 70.8 | 65.3 | 66.3
cRT [38] | 61.8 | 46.2 | 27.3 | 49.6 |  | 69.0 | 66.0 | 63.2 | 65.2
$\tau$-norm [38] | 59.1 | 46.9 | 30.7 | 49.4 |  | 65.6 | 65.3 | 65.5 | 65.6
De-confound [71] | 62.7 | 48.8 | 31.6 | 51.8 |  | — | — | — | —
DiVE [32] | 64.1 | 50.4 | 31.5 | 53.1 |  | 70.6 | 70.0 | 67.6 | 69.1
DRO-LT [65] | 64.0 | 49.8 | 33.1 | 53.5 |  | — | — | — | 69.7
DisAlign [83] | 61.3 | 52.2 | 31.4 | 52.9 |  | 69.0 | 71.1 | 70.2 | 70.6
Our methods (weight balancing)
naive | 55.3 | 31.4 | 12.5 | 38.0 |  | 54.7 | 46.0 | 43.9 | 46.1
WD | 68.5 | 42.4 | 14.2 | 48.6 |  | 74.5 | 66.5 | 61.5 | 65.4
\+ L2norm | 61.2 | 48.9 | 42.6 | 52.8 |  | 11.2 | 47.4 | 66.9 | 51.3
\+ $\tau$-norm | 64.0 | 49.0 | 36.3 | 53.1 |  | 71.3 | 69.8 | 68.9 | 69.6
\+ WD | 62.0 | 49.7 | 41.0 | 53.3 |  | 71.0 | 70.3 | 69.4 | 70.0
\+ Max | 62.2 | 50.1 | 37.5 | 53.0 |  | 71.4 | 68.9 | 69.1 | 69.2
\+ WD & Max | 62.5 | 50.4 | 41.5 | 53.9 |  | 71.2 | 70.4 | 69.7 | 70.2
SOTA with “bells and whistles”: ensembles,
data augmentation, and self-supervised pretraining
RIDE [73] | 67.9 | 52.3 | 36.0 | 56.1 |  | 66.5 | 72.1 | 71.5 | 71.3
ACE [10] | — | — | — | 56.6 |  | — | — | — | 72.9
SSD [46] | 66.8 | 53.1 | 35.4 | 56.0 |  | — | — | — | 71.5
PaCo [17] | 63.2 | 51.6 | 39.2 | 54.4 |  | 69.5 | 72.3 | 73.1 | 72.3

##  5 Conclusion

Long-tailed recognition (LTR) is a crucial challenge for real-world data that
tends to be imbalanced. Our work is motivated by the empirical observation
that a model naively trained over long-tailed data has artificially large
weights for common classes (because they have more data to train than rare
classes). We propose to learn balanced weights via parameter regularization,
including weight decay and MaxNorm regularizers. Our extensive study shows
that properly applying these regularizers greatly boosts LTR performance. We
introduce a simple approach that outperforms prior art on five long-tailed
benchmarks. Because these regularizers are underexplored in the long-tailed
literature, we hope our study draws attention from the practitioners that
parameter regularization should be the first method to consider, when
addressing real-world problems related to the long-tailed distribution.

Limitations. While we focus on the orthogonal direction of parameter
regularization to address LTR, we have not studied how our approaches
complement existing techniques. For example, how to balance weights in
training each of the expert models, or how to balance the weights alongside
sophisticated data augmentation and self-supervised pretraining. We also point
out that other regularization techniques might be better at balancing weights,
for example using L$p$-norm weight decay where $p$$\not=$2 [3]. We leave them
to future work.

Societal Impact. Because real-world data tends to follow long-tailed
distributions, our work has multiple positive societal impacts. For example,
addressing the long-tail proves an important direction for studying bias and
fairness in recognition [15]. However, any system that makes it easier to
train a fair classifier on long-tailed classes also makes it possible for a
malicious agent to train a system that automatically discriminates against a
certain subgroup for which only little training data is available. This is
potentially a negative societal impact.

Acknowledgement. This work was supported by the CMU Argo AI Center for
Autonomous Vehicle Research. SA was supported in part by the KAUST Gifted
Student’s Program (KGSP) and the CMU Robotics Institute Summer Scholars
program. YXW was supported in part by NSF Grant 2106825 and the Jump ARCHES
endowment.

##

Appendix

_In the appendix, we first supplement the ablation study with more results to
justify the use of regularizers for better learning for long-tailed
recognition (LTR). We then present our open-source code in Jupyter Notebook as
a self-explanatory tutorial. Lastly, we attach a video demo that shows how
weights change during training with different regularizers._

##  Appendix A Detailed Ablation Study

In Table 4, we list more results in addition to the ablation study presented
in the main paper. Please refer to the caption for salient conclusions.

##  Appendix B Open-Source Code

Description. We release our code with two executable Jupyter Notebook files
for demonstrating our approaches (w.r.t training and evaluation). The files
will reproduce the results in the ablation study on the CIFAR100-LT dataset
(with an imabalance factor 100). The Jupyter Notebook files are sufficiently
self-explanatory with detailed comments, and displayed output. The first file
compares the first-stage training between naive training and training with
weight decay. The second file studies different regularizers in the second
stage training. We advise the reader to run the files in order (if running
them) because the second stage training (i.e., the second demo file) requires
the saved model by the first file. Running the first file takes $\sim$2 hours
with a GPU (NVIDIA GeForce RTX 3090), and the second file takes a few minutes.

  * •

        demo1_first-stage-training.ipynb

Running this file compares the first-stage training between a naive network
(without weight decay) and a model with a tuned weight decay. It should
achieve an overall accuracy $\sim$39% and $\sim$46% respectively on the
CIFAR100-LT (imbalance factor 100).

  * •

        demo2_second-stage-training.ipynb

Running this file will compare various regularizers used in the second-stage
training. It should achieve an overall accuracy $>$52%.

Why Jupyter Notebook? We prefer to release the code using Jupyter Notebook
(https://jupyter.org) because it allows for interactive demonstration for
education purposes. In case the reader would like to run python script, using
the following command can convert a Jupyter Notebook file XXX.ipynb into a
Python script:

    jupyter nbconvert --to script XXX.ipynb

Requirement. Running our code requires some common packages. We installed
Python and most packages through Anaconda. A few other packages might not be
installed automatically, such as Pandas, torchvision, and PyTorch, which are
required to run our code. Below are the versions of Python and PyTorch used in
our work.

  * •

Python version: 3.7.4 [GCC 7.3.0]

  * •

PyTorch verion: 1.7.1

We suggest assigning $>$1GB space to run all the files. The code will save
checkpoints after every training epoch.

License. We release open-source code under the MIT License to foster future
research in this field.

Table 4: Ablation study on CIFAR100-LT (IF=100) w.r.t top-1 accuracy (%). “CE”: cross-entropy loss; “CB”: class-balanced loss [19]; “WD”: weight decay; “Max”: MaxNorm constraint; “default-WD”: using the weight decay tuned for the first-stage training; “$\tau$-norm”: $\tau$-normalization [38]; “+”: fine-tuning the last layer(s) as the second-stage training. Here are salient conclusions. (1) Learning with a properly tuned WD boosts performance from 38.38% to 46.08%, that is +8% increase. (2) Re-training the last layer with CB and WD gives another boost (+6%) to 52.42%. (3) Based on the above, applying additional MaxNorm yields a slight improvement +1% (53.35%); finetuning the last two layers achieves 53.55%. (4) Finetuning more layers performs worse, presumably because CB induces modified gradients that affect feature learning, and so hurt the final LTR performance.  model | many | median | few | avg
---|---|---|---|---
on the last layer (classifier)
WD=0 (w/ CE) | 64.05 | 35.80 | 11.43 | 38.38
\+ $\tau$-norm ($\tau$ =1.0) | 59.54 | 38.23 | 25.93 | 42.00
WD tuned (w/ CE) | 76.94 | 44.28 | 12.17 | 46.08
\+ $\tau$-norm ($\tau$ =1.9) | 73.11 | 47.69 | 30.10 | 51.31
\+ L2norm | 76.09 | 47.74 | 20.87 | 49.60
\+ CE & L2norm | 76.37 | 48.11 | 21.00 | 49.87
\+ CE & WD | 76.97 | 45.94 | 14.00 | 47.22
\+ CE & Max | 76.80 | 47.26 | 15.10 | 47.95
\+ CE & Max & default-WD | 76.89 | 47.06 | 13.90 | 47.55
\+ CE & Max & WD | 76.80 | 47.51 | 14.40 | 47.83
\+ CB | 77.00 | 45.89 | 13.60 | 47.09
\+ CB & L2norm | 76.43 | 48.20 | 21.60 | 50.10
\+ CB & WD | 72.77 | 49.74 | 31.80 | 52.42
\+ CB & Max | 76.49 | 49.23 | 20.67 | 50.20
\+ CB & Max & default-WD | 76.20 | 48.91 | 21.50 | 50.24
\+ CB & WD & Max | 72.60 | 51.86 | 32.63 | 53.35
on the last two layers
\+ CE & WD & Max | 76.34 | 48.46 | 21.17 | 50.03
\+ CB & WD & Max | 71.37 | 51.17 | 35.53 | 53.55
on the last five layers
\+ CE & WD & Max | 76.03 | 48.14 | 20.87 | 49.72
\+ CB & WD & Max | 74.37 | 49.80 | 26.63 | 51.45

##  Appendix C Video Demo

(a) Naively trained network without weight decay

(b) Network trained with L2-normalization

(c) Network trained with weight decay

(d) MaxNorm constrained network

Figure 7: We plot per-class filter weights of the classifier as 2D points from
(a) a naively trained network without weight decay, (b) a classifier with
L2-normalization, (c) a classifier with weight decay, and (d) a classifier
constrained by MaxNorm. All the networks are trained on the CIFAR100-LT
dataset with imbalance factor as 100. The four columns denote training
iterations: iteration-0 as random initialization, iteration-4000,
iteration-8000, and iteration-16000. The naively trained network learns
“imbalanced” weights, i.e., large weights for the common classes and small
weights for the rare classes. The model trained with L2-normalization has
constant weight norms. When trained with weight decay, the network has smaller
yet more balanced weights for all the classes. The MaxNorm constrained network
caps weight norms, encouraging small weights (from both common and rare
classes) to grow, approaching to the surface of the norm ball. We refer the
reader to the video demo demo2D_weight_evolution.mp4 for better visualization.

The goal of this section is to demonstrate how weights’ norms evolve during
training. For demonstration, we train models on the CIFAR100-LT dataset with
an imbalance factor 100. To do so, we modify the ResNet34 network architecture
by inserting an additional 2-dim pre-logit layer. This layer has weights ${\bf
W}=[{\bf w}_{ij}]\in{\mathbb{R}}^{2\times K}$ that project 2-dim pre-logit
features to $K$-dim logits. At the logit layer, each filter weight ${\bf
w}_{i}$ (i.e., a row of ${\bf W}$) is class-specific. Therefore, we can plot
the $K$ 2-dim class-specific weights as $K$ points on a 2D plane. The MaxNorm
constraint upper bounds the norm of each class-specific weight, i.e., $\|{\bf
w}_{i}\|_{2}<\delta$. Fig. 7 plots per-class weights after three different
training iterations. For better visualization, we suggest the reader to watch
our video demo demo2D_weight_evolution.mp4 in our github repository
https://github.com/ShadeAlsha/LTR-weight-
balancing/blob/master/demo2D_weight_evolution.mp4.

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# Explicit $K_{2}$ of some quotient rings of $\mathbb{Z}[G]$, $G$ an
elementary abelian $p$-group

Yakun Zhang<EMAIL_ADDRESS>School of Mathematics, Nanjing Audit
University, Nanjing 211815, China

###### Abstract

Let $G$ be an elementary abelian $p$-group. We calculate
$K_{2}(\mathbb{Z}[G]/I)$ for certain ideals $I\subseteq\mathbb{Z}[G]$ of the
finite $p$-power index through computations of Dennis-Stein symbols. As an
application, for $p$ an odd prime, we give a complete description of the
relative group $SK_{1}(\mathbb{Z}[G],p^{k}\mathbb{Z}[G])$ and obtain more
precise lower bounds of the $p$-rank of $K_{2}(\mathbb{Z}[G])$ and
$Wh_{2}(G)$.

###### keywords:

$K_{2}$-groups , Finite group algebras , Dennis-Stein symbols

###### MSC:

[2010] 16S34; 19C20; 19C99

††journal: XXXX

##  1 Introduction

Let $G$ be a finite abelian group of order $p^{n}$, $\Gamma$ the maximal
$\mathbb{Z}$-order of $\mathbb{Z}[G]$. By Proposition 2.2 in [2], $\Gamma$ is
isomorphic to some direct product of ring of algebraic integers, and
$p^{n}\Gamma\subseteq\mathbb{Z}[G]\subseteq\Gamma$. In [4], H.Chen, Y. Gao,
and G.Tang obtained an explicit description of $\Gamma$ (see [4, Lemma 3.3])
and a lower bound of the order of $K_{2}(\mathbb{Z}[G]/p^{n}\Gamma)$, where
$G$ is an elementary abelian $p$-group or a cyclic $p$-group, but the explicit
structure of $K_{2}(\mathbb{Z}[G]/p^{n}\Gamma)$ is still a mystery. Since
$K_{2}(\mathbb{Z}[G]/p^{n}\Gamma)$ sits in the Mayer-Vietoris sequence:

$K_{2}(\mathbb{Z}[G]){\rightarrow}K_{2}(\mathbb{Z}[G]/p^{r}\Gamma)\oplus
K_{2}(\Gamma){\rightarrow}K_{2}(\Gamma/p^{n}\Gamma){\rightarrow}SK_{1}(\mathbb{Z}[G]){\rightarrow}1,$
(1.1)

one motivation of their research is that $K_{2}(\mathbb{Z}[G]/p^{n}\Gamma)$
can be used to give a lower bound of the order of $K_{2}(\mathbb{Z}[G])$,
which has important applications in both topology and algebra. Inspired by
their work, we will in this paper focus on computing $K_{2}$ of some quotient
rings of $\mathbb{Z}[G]$. For any $k\geq n$, the relationship between
$\mathbb{Z}[G]$ and $\mathbb{Z}[G]/p^{k}\Gamma$ can be described by the
following Cartesian square of rings:

$\begin{CD}\mathbb{Z}[G]@>{}>{}>\Gamma\\\ @V{}V{}V@V{}V{}V\\\
\mathbb{Z}[G]/p^{k}\Gamma @>{}>{}>\Gamma/p^{k}\Gamma\end{CD}$ (1.2)

by [2, Theorem 1.3], the square induces a Mayer-Vietoris sequence which is
parallel to (1.1):

$K_{2}(\mathbb{Z}[G]){\rightarrow}K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)\oplus
K_{2}(\Gamma){\rightarrow}K_{2}(\Gamma/p^{k}\Gamma){\rightarrow}SK_{1}(\mathbb{Z}[G]){\rightarrow}1.$
(1.3)

Therefore, $K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$ carries more weight than
$K_{2}(\mathbb{Z}[G]/p^{n}\Gamma)$ to estimate the lower bounds of the order
and the $p$-rank of $K_{2}(\mathbb{Z}[G])$, however, due to the complex
structure of $\Gamma$, it is hard to determinate the explicit structure of
$K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$ directly, in [12], we investigate
$K_{2}(\mathbb{Z}[G])$ through studying the inver-limit of (1.3) to avoid this
difficult problem. For simplicity, in this paper we mainly consider the case
$G$ an elementary abelian $p$-group of rank $n$.

Recall that the pro-$p$-group $K_{2}^{c}(\widehat{\mathbb{Z}}_{p}[G])$ is
defined to be the inverse limit of the groups $K_{2}(\mathbb{Z}[G]/I)$ taken
over all ideals $I\subseteq\mathbb{Z}[G]$ of finite $p$-power index (see [8]).
According to [8, Lemma 3.8], when $G$ is an elementary abelian $p$-group of
rank $n$, we have

$K_{2}^{c}(\widehat{\mathbb{Z}}_{p}[G])\cong
K_{2}^{c}(\widehat{\mathbb{Z}}_{p})\oplus
HC_{1}(\widehat{\mathbb{Z}}_{p}[G]),$

where $HC_{1}(\widehat{\mathbb{Z}}_{p}[G])$ means the first cyclic homology of
$\widehat{\mathbb{Z}}_{p}[G]$ (see [8, page 532]), by definition, we have

$HC_{1}(\widehat{\mathbb{Z}}_{p}[G])\cong(G\otimes\widehat{\mathbb{Z}}_{p}[G])/\langle
g\otimes\lambda g:g\in
G,\lambda\in\widehat{\mathbb{Z}}_{p}\rangle\cong\prod_{g\in G}G/\langle
g\rangle,$

thus the order of $K_{2}^{c}(\widehat{\mathbb{Z}}_{p}[G])$ is closely related
to the number of cyclic subgroups of $G$:

$|K_{2}^{c}(\widehat{\mathbb{Z}}_{p}[G])|=|K_{2}^{c}(\widehat{\mathbb{Z}}_{p})|\cdot\prod_{g\in
G}|G/\langle g\rangle|.$ (1.4)

By definition,

$K_{2}^{c}(\widehat{\mathbb{Z}}_{p}[G])\cong\lim\limits_{\overset{\longleftarrow}{k}}K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)\cong\lim\limits_{\overset{\longleftarrow}{k}}K_{2}(\mathbb{Z}[G]/(p^{k})),$

in which $K_{2}(\mathbb{Z}[G]/(p^{k}))$ was firstly investigated in [1, 2] to
compute $SK_{1}(\mathbb{Z}[G])$ and is more accessible to computation.
According to [2, Proposition 1.1] (see Lemma 2.1), surjective homomorphism
between two commutative semi-local rings induces a surjection on $K_{2}$, thus
(1.4) gives the upper bound of the order of $K_{2}(\mathbb{Z}[G]/(p^{k}))$ and
$K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$. Then three questions naturally arise:

  1. (i)

What are the generators of $K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$ and
$K_{2}(\mathbb{Z}[G]/(p^{k}))$ ?

  2. (ii)

Under what conditions are $K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$ and
$K_{2}(\mathbb{Z}[G]/(p^{k}))$ stable ?

  3. (iii)

Under what conditions are $K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$ and
$K_{2}(\mathbb{Z}[G]/(p^{k}))$ isomorphic ?

To give a complete answer to the above questions, the article is organized as
follows: firstly, we give explicit structure of $K_{2}(\mathbb{F}_{p}[G])$ in
terms of Dennis-Stein symbols or Steinberg symbols (see Lemma 2.3), then apply
this result to give the explicit structure of $K_{2}(\mathbb{Z}[G]/(p^{k}))$
(see Theorem 2.6), on this basis, we give a complete description of
$K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$ (see Theorem 3.4), our results show that
$K_{2}(\mathbb{Z}[G]/(p^{k}))$ is stable for $k\geq 2$,
$K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$ is stable for $k\geq n+1$. Since the two
$K_{2}$-groups share the same inverse limit, $K_{2}(\mathbb{Z}[G]/(p^{k}))$
and $K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$ are isomorphic for $k\geq n+1$. As an
application, for $p$ an odd prime, we compute $SK_{1}(\mathbb{Z}[G],(p^{k}))$
for all $k\geq 1$, and give new lower bounds of the $p$-rank of
$K_{2}(\mathbb{Z}[G])$ and $Wh_{2}(G)$, which improves conclusions of [5], the
results are consistent with our previous estimations (see [12, Theorem 2.7])
in most case.

Notations

When $a$ and $b$ are commutative elements of a ring $R$ with units satisfying
$1-ab\in R^{*}$, the Dennis-Stein symbol $\langle a,b\rangle$ lies in
$K_{2}(R)$, here we adopt the “symmetric” Dennis-Stein symbol $\langle
a,b\rangle$ defined in [11], thus $\langle a,b\rangle$ here corresponds to
$\langle-a,b\rangle$ in [1] and [9]. The relative $K_{2}$-group $K_{2}(R,I)$
is generated by the Dennis-Stein symbols $\langle a,b\rangle$, with $a$ or $b$
lies in an ideal $I$ of $R$. Dennis-Stein symbols subject to the following
relations:

$\displaystyle(\mbox{DS1})\quad$ $\displaystyle\langle a,b\rangle\langle
b,a\rangle=1;$ $\displaystyle(\mbox{DS2})\quad$ $\displaystyle\langle
a,b\rangle\langle a,c\rangle=\langle a,b+c-abc\rangle;$
$\displaystyle(\mbox{DS3})\quad$ $\displaystyle\langle a,bc\rangle=\langle
ab,c\rangle\langle ac,b\rangle.$

If $a\in R^{*}$, then $\langle a,b\rangle=\\{a,1-ab\\}$. $C_{n}$ will always
denote a multiplicative cyclic group of order $n$, and $G$ will always denote
an elementary abelian $p$-group of rank $n$. Set
$G=\langle\sigma_{i}\mid\sigma_{i}^{p}=1,\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},1\leq
i,j\leq n\rangle$, $x_{i}=\sigma_{i}-1,$ $1\leq i\leq n$, and define

$\displaystyle T=\\{\langle
x_{i},\prod_{j=1}^{n}x_{j}^{\lambda_{j}}\rangle|1\leq i\leq
n,0\leq\lambda_{j}\leq p-1,\sum_{j=1}^{n}\lambda_{j}>0\\},$ $\displaystyle
T_{1}=\\{\langle x_{i},\prod_{j=i}^{n}x_{j}^{\lambda_{j}}\rangle|1\leq i\leq
n,0\leq\lambda_{i}\leq p-2,0\leq\lambda_{i+1},\cdots,\lambda_{n}\leq
p-1,\sum_{j=i}^{n}\lambda_{j}>0\\},$ $\displaystyle T_{2}=\\{\langle
x_{i},x_{i}^{p-1}\rangle|1\leq i\leq n\\},T_{3}=\\{\langle
x_{i},\prod_{j=1}^{n}x_{j}^{p-1}\rangle|1\leq i\leq n\\}.$

Finally, for any prime $p$, we define

$\tau(p)=\left\\{\begin{array}[]{ccl}1,&&{\mbox{if }p=2,}\\\ 0,&&{\mbox{if
}p\neq 2.}\end{array}\right.$

##  2 The explicit structure of $K_{2}(\mathbb{Z}[G]/(p^{k}))$ for $k\geq 2$

######  Lemma 2.1.

[2, Proposition 1.1] Let $f:A\rightarrow B$ be a surjective homomorphism
between two commutative semi-local rings. Then $f$ induces a surjection
$K_{2}(A)\rightarrow K_{2}(B)$.

######  Lemma 2.2.

Let $x_{1},\cdots,x_{n}$ be elements of a ring such that $1-x$ is invertible,
where $x=\prod_{i=1}^{n}x_{i}$. Then

$\prod\limits_{i=1}^{n}\langle x_{i},x/x_{i}\rangle=1.$

######  Proof.

The Lemma is proved by iteration of the Dennis-Stein symbols product rules
$(\mbox{DS1})$ and $(\mbox{DS3})$. ∎

######  Lemma 2.3.

$K_{2}(\mathbb{F}_{p}[G])$ is an elementary abelian $p$-group of rank
$(n-1)(p^{n}-1)$. Set $T\backslash(T_{1}\cup T_{2})$ forms a basis for
$K_{2}(\mathbb{F}_{p}[G])$.

######  Proof.

According to Proposition 3 and Proposition 5 in [5],
$K_{2}(\mathbb{F}_{p}[G])$ is an elementary abelian $p$-group, the $p$-rank of
$K_{2}(\mathbb{F}_{p}[G])$ is exactly $(n-1)(p^{n}-1)$. Set

$S=\\{\prod_{j=1}^{n}x_{j}^{\lambda_{j}}|0\leq\lambda_{j}\leq
p-1,\sum_{j=1}^{n}\lambda_{j}>0\\},\quad
S_{i}=\\{x_{i}^{\lambda_{i}}|0<\lambda_{i}\leq p-1\\}.$

then by Lemma 2.2, for $s\in S_{i},$ $\langle x_{i},s\rangle$ is trivial. Note
that $\langle-p,-p\rangle=0$, $x_{j}^{p}=0$, and Proposition 1.1 in [1] also
holds for $k=1$ with slight modifications, then $K_{2}(\mathbb{F}_{p}[G])$ is
generated by

  1. (i)

all Dennis-Stein symbols $\langle x_{i},s_{i}\rangle,s_{i}\in S\backslash
S_{i},1\leq i\leq n$, or equivalently, by

  2. (ii)

all Steinberg symbols $\\{1+x_{i},1+s_{i}\\},s_{i}\in S\backslash S_{i},1\leq
i\leq n.$

The number of Dennis-Stein symbols listed here is $n(p^{n}-p)$. For a fixed
number $t$, $2\leq t\leq n$, write
$\widetilde{x}_{l_{j}}=(\prod_{j=1}^{t}x_{l_{j}}^{\lambda_{l_{j}}})/x_{l_{j}}$,
where $1\leq j\leq t$, $\widetilde{x}_{l_{j}}\neq 1$,
$1\leq\lambda_{l_{j}}\leq p-1$, $1\leq l_{j}\leq n$, and the $l_{j}$’s are
arranged by size of subscript. The number of generators as $\langle
x_{l_{1}},\widetilde{x}_{l_{1}}\rangle$ listed above is
$\dbinom{n}{t}(p-1)^{t}$. By Lemma 2.2, $\langle
x_{l_{j}},\widetilde{x}_{l_{j}}\rangle$ ($1\leq j\leq t$) are linearly
dependent:

$\prod_{j=1}^{t}\langle
x_{l_{j}},\widetilde{x}_{l_{j}}\rangle^{\lambda_{j}}=1,$

thus $K_{2}(\mathbb{F}_{p}[G])$ has $p$-rank at most

$n(p^{n}-p)-\sum\limits_{i=2}^{n}\binom{n}{i}(p-1)^{i}=(n-1)(p^{n}-1),$

which coincides with the previous estimate, it follows that the set
$T\backslash(T_{1}\cup T_{2})$ forms a basis for $K_{2}(\mathbb{F}_{p}[G])$. ∎

######  Lemma 2.4.

Let $\sigma$ be one generator of $C_{p}$, $k\geq 2$. For $p=2$,
$K_{2}(\mathbb{Z}[C_{p}]/(p^{k}))$ is an elementary abelian 2-group of rank 2
generated by $\langle\sigma-1,\sigma-1\rangle$ and $\langle-2,-2\rangle$. For
$p$ an odd prime, $K_{2}(\mathbb{Z}[C_{p}]/(p^{k}))$ is a cyclic group of
order $p$ generated by $\langle\sigma-1,(\sigma-1)^{p-1}\rangle$.

######  Proof.

For $p=2$, $SK_{1}(\mathbb{Z}[C_{2}],(2^{k}))$ is trivial [1, Theorem
1.10(c)], $K_{2}(\mathbb{Z}[C_{2}]/(2^{k}))\cong K_{2}(\mathbb{Z}[C_{2}]/(4))$
is an elementary abelian 2-group of rank 2 [smallK2, Theorem 2.1], then by the
algebraic $K$-theory exact sequence associated to the pair
$(\mathbb{Z}[C_{2}],2^{k}\mathbb{Z}[C_{2}])$, $K_{2}(\mathbb{Z}[C_{2}])$ maps
isomorphically onto $K_{2}(\mathbb{Z}[C_{2}]/(2^{k}))$, it is well known that
$K_{2}(\mathbb{Z}[C_{2}])$ is generated by $\\{\sigma,\sigma\\}$ and
$\\{-1,-1\\}$ [10], note that $(\sigma-1)^{2}=2(\sigma-1)$, and
$K_{2}(\mathbb{Z}[C_{2}]/(4))$ has exponent 2, then by Proposition 1.1 in [1],
$K_{2}(\mathbb{Z}[C_{2}]/(4))$ is generated by
$\langle\sigma-1,\sigma-1\rangle$ and $\langle-2,-2\rangle$.

For $p$ an odd prime, by (1.4), the inverse limit of
$K_{2}(\mathbb{Z}[C_{p}]/(p^{k}))$ is a cyclic group of order $p$, to prove
our result, it suffices to prove $K_{2}(\mathbb{Z}[C_{p}]/(p^{k}))$ is a
cyclic $p$-group of order $p$ generated by
$\langle\sigma-1,(\sigma-1)^{p-1}\rangle$. Let $P$ be the maximal ideal of
$A=\mathbb{Z}[\zeta_{p}]$ generated by $\pi=\zeta_{p}-1$, then by algebraic
number theory, $pA={\pi}^{p-1}A$. According to Theorem 4.3 in [6], for any
$k\geq 2$, $K_{2}(A/(p^{k}))$ is isomorphic to $K_{2}(A/P^{p})$, which is a
cyclic group of order $p$, while according to Corollary 3.3 and the exact
sequence (3.1) in the page 209 of [6], $K_{2}(A/P^{p})$ is isomorphic to
$K_{2}(A/P^{p},P^{p-1}/P^{p})$, which is generated by
$\\{1+\pi,1+{\pi}^{p-1}\\}$ by [6, Lemma 3.2], note that ${\pi}^{p}=0$ holds
in $A/P^{p}$, then

$\langle\pi,{\pi}^{p-1}\rangle=\langle\pi,{\pi}^{p-1}\rangle\langle
1,{\pi}^{p-1}\rangle=\langle 1+\pi,{\pi}^{p-1}\rangle=\langle
1+\pi,-{\pi}^{p-1}\rangle^{-1}=\\{1+\pi,1+{\pi}^{p-1}\\}^{-1},$

thus $K_{2}(A/P^{p},P^{p-1}/P^{p})$ is also generated by
$\langle\pi,{\pi}^{p-1}\rangle$, it follows that for any $k\geq 2$,
$K_{2}(A/(p^{k}))$ is generated by $\langle\pi,{\pi}^{p-1}\rangle$. According
to [2, Theorem 2.10], the homomorphism
$\chi:K_{2}(\mathbb{Z}[C_{p}]/(p^{k}))\rightarrow K_{2}(A/(p^{k}))$ induced by
canonical map of rings is surjective, obviously
$\langle\sigma-1,(\sigma-1)^{p-1}\rangle$ maps isomorphically onto
$\langle\pi,{\pi}^{p-1}\rangle$, then our result follows. ∎

######  Remark 2.5.

There exists another method to prove $\\{1+\pi,1+{\pi}^{p-1}\\}$ is non-
trivial, see [roberts2006k2, pages 269-270] for more details. From the proof
of Lemma 2.4, we conclude that
$\langle\pi,{\pi}^{p-1}\rangle=\\{1+{\pi}^{p-1},1+\pi\\}$, and that in
$K_{2}(\mathbb{Z}[C_{p}]/(p^{k}))$ we have
$\langle\sigma-1,(\sigma-1)^{p-1}\rangle=\\{1+(\sigma-1)^{p-1},\sigma\\}$.

######  Theorem 2.6.

For any $k\geq 2$, $K_{2}(\mathbb{Z}[G]/(p^{k}))$ is an elementary abelian
$p$-group of rank ${(n-1)p^{n}+1+\tau(p)}$. The set $T\backslash T_{1}$
together with $\langle-p,-p\rangle$ (if $p$=2) forms a basis for
$K_{2}(\mathbb{Z}[G]/(p^{k}))$.

######  Proof.

By (1.4), the inverse limit of $K_{2}(\mathbb{Z}[G]/(p^{k}))$ has $p$-rank
$(n-1)p^{n}+1+\tau(p)$, thus the $p$-rank of $K_{2}(\mathbb{Z}[G]/(p^{k}))$ is
at most $(n-1)p^{n}+1+\tau(p).$ On the other hand, set
$G_{i}=\langle\sigma_{i}\rangle$, $1\leq i\leq n$, then
$\mathbb{F}_{p}[G_{i}]\cong\mathbb{F}_{p}[x_{i}]/({x_{i}}^{p})$, by
Corollary4.4(a) in [6], $K_{2}(\mathbb{F}_{p}[x_{i}]/({x_{i}}^{p}))$ is
trivial. By Lemma 2.1, for any $k\geq 2$, $K_{2}(\mathbb{Z}[G]/(p^{k}))$ maps
onto $K_{2}(\mathbb{F}_{p}[G])$, which is an elementary abelian $p$-group of
rank $(n-1)(p^{n}-1)$ by Lemma 2.3. Since $K_{2}(\mathbb{F}_{p}[G_{i}])$ is
trivial, the kernel contains $n+\tau(p)$ elements: $\langle-p,-p\rangle$ (if
$p=2$) and $\langle x_{i},x_{i}^{p-1}\rangle$, $1\leq i\leq n$, the
$n+\tau(p)$ elements are obviously linearly independent. According to the
proof of Lemma 2.4, $\langle x_{i},x_{i}^{p-1}\rangle$ maps isomorphically
onto the generator of $K_{2}(\mathbb{Z}[G_{i}]/(p^{k}))$, it follows that the
$p$-rank of $K_{2}(\mathbb{Z}[G]/(p^{k}))$ is at least $(n-1)p^{n}+1+\tau(p)$,
which completes the proof. ∎

##  3 The explicit structure of $K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$ for $k\geq
n$

Let $G$ be an elementary abelian $p$-group of rank $n\geq 2$, $\Gamma$ the
maximal $\mathbb{Z}$-order of $\mathbb{Z}[G]$, then according to [4, Lemma
3.3], $\Gamma=\sum_{H\leq G}\mathbb{Z}[G]e_{H},$ where $e_{H}=\sum_{h\in
H}h/|H|$, $e_{H}$ is idempotent. Set $J=|G|\Gamma$, $\widetilde{G}=\sum_{g\in
G}g$, and let $I$ be the ideal of $\mathbb{Z}[G]$ generated by
$\\{|G|e_{H}|H\leq G,H\neq G\\}$. Then $J=I+(\widetilde{G})\mathbb{Z}$, note
that $I\subseteq p\mathbb{Z}[G]$, hence $p(\widetilde{G})\mathbb{Z}\subseteq
J\cap p\mathbb{Z}[G]=I$, it follows that $pJ\subseteq I\subseteq J$, and that
there is a Cartesian square

$\begin{CD}\mathbb{Z}[G]/(p\mathbb{Z}[G]\cap J)@>{}>{}>\mathbb{Z}[G]/J\\\
@V{}V{}V@V{}V{}V\\\
\mathbb{Z}[G]/p\mathbb{Z}[G]@>{}>{}>\mathbb{Z}[G]/(p\mathbb{Z}[G]+J)\end{CD}\quad\mbox{i.e.,
}\quad\begin{CD}\mathbb{Z}[G]/I@>{}>{}>\mathbb{Z}[G]/J\\\ @V{}V{}V@V{}V{}V\\\
\mathbb{F}_{p}[G]@>{}>{}>\mathbb{F}_{p}[G]/(\widetilde{G})\end{CD}$ (3.1)

In [4], the square (3.1) has been used to give a lower bound of the order of
$K_{2}(\mathbb{Z}[G]/|G|\Gamma)$. In this section, we turn to the explicit
calculation of $K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$ for any $k\geq n$ by making
a subtle analysis of the $K_{2}$-groups of finite group rings lying in the
square (3.1). For proving Theorem 3.4, some technical lemmas will be needed.
With a minor modification of the proof of Lemma 2.4, we obtain the first
lemma:

######  Lemma 3.1.

Let $\sigma$ be one generator of $C_{p}$, $\Gamma$ the maximal
$\mathbb{Z}$-order of $\mathbb{Z}[C_{p}]$, $k\geq 2$, For $p=2$,
$K_{2}(\mathbb{Z}[C_{p}]/p^{k}\Gamma)$ is an elementary abelian 2-group of
rank 2 generated by $\langle\sigma-1,\sigma-1\rangle$ and $\langle
2,2\rangle$. For $p$ an odd prime, $K_{2}(\mathbb{Z}[C_{p}]/p^{k}\Gamma)$ is a
cyclic group of order $p$ generated by
$\langle\sigma-1,(\sigma-1)^{p-1}\rangle$.

######  Lemma 3.2.

Let $G$ be an elementary abelian $p$-group of rank $n\geq 2$. Then
$K_{2}(\mathbb{F}_{p}[G]/(\widetilde{G}))$ is an elementary abelian $p$-group
of rank $(n-1)(p^{n}-2)-1$, set $T\backslash(T_{1}\cup T_{2}\cup T_{3})$ forms
a basis for $K_{2}(\mathbb{F}_{p}[G]/(\widetilde{G}))$.

######  Proof.

Let
$G=\langle\sigma_{i}\mid\sigma_{i}^{p}=1,\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},1\leq
i,j\leq n\rangle$. By Lemma 4.1 in [4],
$K_{2}(\mathbb{F}_{p}[G],(\widetilde{G}))$ is an elementary abelian $p$-group
of rank $n$ with generators $\langle\sigma_{i},\widetilde{G}\rangle$, $1\leq
i\leq n$. Note that $(\sigma_{i}-1)\widetilde{G}=0$, and

$\widetilde{G}=\prod_{j=1}^{n}(\sum_{i=0}^{p-1}(\sigma_{j})^{i})\equiv\prod_{j=1}^{n}(\sigma_{j}-1)^{p-1}\pmod{p\mathbb{Z}[G]}.$

It follows that

$\langle\sigma_{i},\widetilde{G}\rangle=\langle\sigma_{i}-1,\widetilde{G}\rangle\langle
1,\widetilde{G}\rangle=\langle\sigma_{i}-1,\widetilde{G}\rangle=\langle\sigma_{i}-1,\prod_{j=1}^{n}(\sigma_{j}-1)^{p-1}\rangle,$

then $K_{2}(\mathbb{F}_{p}[G],(\widetilde{G}))$ is a direct summand of
$K_{2}(\mathbb{F}_{p}[G])$. Since $K_{2}(\mathbb{F}_{p}[G])$ has exponent $p$,
the following exact sequence splits:

$1\rightarrow K_{2}(\mathbb{F}_{p}[G],(\widetilde{G}))\rightarrow
K_{2}(\mathbb{F}_{p}[G])\rightarrow
K_{2}(\mathbb{F}_{p}[G]/(\widetilde{G}))\rightarrow 1,$ (3.2)

then the result follows from Lemma 2.3. ∎

######  Lemma 3.3.

Let $G$ be an elementary abelian $p$-group of rank $n\geq 2$. Then
$K_{2}(\mathbb{Z}[G]/I)$ is an elementary abelian $p$-group of rank
$(n-1)p^{n}+1+\tau(p)$, set $T\backslash T_{1}$ together with
$\langle-p,-p\rangle$ (if $p$=2) forms a basis for $K_{2}(\mathbb{Z}[G]/I)$.

######  Proof.

Let
$G=\langle\sigma_{i}\mid\sigma_{i}^{p}=1,\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},1\leq
i,j\leq n\rangle$, and $G_{i}=\langle\sigma_{i}\mid\sigma_{i}^{p}=1\rangle$,
and let $\Gamma_{i}$ be the maximal $\mathbb{Z}$-order of $\mathbb{Z}[G_{i}]$,
$1\leq i\leq n$. According to [4, page 1587], the canonical map of rings
$f:\mathbb{Q}[G]\rightarrow\mathbb{Q}[G_{i}]$ gives rise to
$f(|G|\Gamma)=|G|\Gamma_{i}$, and induces a surjective map of rings
$\mathbb{Z}[G]/|G|\Gamma\rightarrow\mathbb{Z}[G_{i}]/|G|\Gamma_{i}$, so, the
map $g:K_{2}(\mathbb{Z}[G]/I)\rightarrow K_{2}(\mathbb{F}_{p}[G])$ is
surjective by Lemma 2.1. $\mbox{Ker}(g)$ contains
$\langle\sigma_{i}-1,(\sigma_{i}-1)^{p-1}\rangle$ ($1\leq i\leq n$) and
$\langle-p,-p\rangle$ (if $p=2$), these generators have exponent $p$ by Lemma
3.1. Since $p^{n+1}\mathbb{Z}[G]\subseteq pJ\subseteq I\subseteq
p\mathbb{Z}[G]$, we conclude from the proof of Theorem 2.6 that

$\mbox{Ker}(K_{2}(\mathbb{Z}[G]/(p^{n+1}))\rightarrow
K_{2}(\mathbb{F}_{p}[G]))\cong\mbox{Ker}(K_{2}(\mathbb{Z}[G]/I)\rightarrow
K_{2}(\mathbb{F}_{p}[G])),$

then $K_{2}(\mathbb{Z}[G]/I)$ is isomorphic to
$K_{2}(\mathbb{Z}[G]/(p^{n+1}))$, the result follows. ∎

######  Theorem 3.4.

Let $G$ be an elementary abelian $p$-group of rank $n\geq 2$. Then
$K_{2}(\mathbb{Z}[G]/p^{n}\Gamma)$ is an elementary abelian $p$-group of rank
$(n-1)(p^{n}-1)+\tau(p)$, the set $T\backslash(T_{1}\cup T_{3})$ together with
$\langle-p,-p\rangle$ (if $p$=2) forms a basis for
$K_{2}(\mathbb{Z}[G]/p^{n}\Gamma)$. For any $k\geq n+1$,
$K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$ is an elementary abelian $p$-group of rank
$(n-1)p^{n}+1+\tau(p)$, the set $T\backslash T_{1}$ together with
$\langle-p,-p\rangle$ (if $p$=2) forms a basis for
$K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$.

######  Proof.

The Cartesian square (3.1) gives rise to a natural commutative diagram with
exact rows:

$\setcounter{MaxMatrixCols}{11}\begin{CD}K_{2}(\mathbb{Z}[G]/I,J/I)@>{f_{1}}>{}>K_{2}(\mathbb{Z}[G]/I)@>{f_{2}}>{}>K_{2}(\mathbb{Z}[G]/J)@>{}>{}>1\\\
@V{g_{1}}V{}V@V{g_{2}}V{}V@V{g_{3}}V{}V\\\
1@>{}>{}>K_{2}(\mathbb{F}_{p}[G],(\widetilde{G}))@>{}>{}>K_{2}(\mathbb{F}_{p}[G])@>{}>{}>K_{2}(\mathbb{F}_{p}[G]/(\widetilde{G}))@>{}>{}>1\end{CD}$

where the bottom row is given by (3.2). By Lemma 2.1, $f_{2}$, $g_{2}$ and
$g_{3}$ are surjective. By Lemma 4.2 in [4], $g_{1}$ is an isomorphism, then
$K_{2}(\mathbb{Z}[G]/I,J/I)$ is an elementary abelian $p$-group and $f_{1}$ is
injective. A corollary of Lemma 2.1 and Theorem 2.6 shows that both
$K_{2}(\mathbb{Z}[G]/I)$ and $K_{2}(\mathbb{Z}[G]/J)$ are elementary abelian
$p$-group, it follows that the top exact sequence splits. By Snake Lemma,
$\mbox{Ker}(g_{2})\cong\mbox{Ker}(g_{3})$, also, note that both $g_{2}$ and
$g_{3}$ are split surjections, then

$K_{2}(\mathbb{Z}[G]/J)\cong
K_{2}(\mathbb{F}_{p}[G]/(\widetilde{G}))\oplus\mbox{Ker}\big{(}K_{2}(\mathbb{Z}[G]/I)\rightarrow
K_{2}(\mathbb{F}_{p}[G])\big{)}.$

By Lemma 2.3, $K_{2}(\mathbb{F}_{p}[G])$ has $p$-rank $(n-1)(p^{n}-1)$, by
Lemma 3.2, $K_{2}(\mathbb{F}_{p}[G]/(\widetilde{G}))$ has $p$-rank
$(n-1)(p^{n}-2)-1$, by Lemma 3.3, $K_{2}(\mathbb{Z}[G]/I)$ has $p$-rank
$(n-1)p^{n}+1+\tau(p)$. Then $K_{2}(\mathbb{Z}[G]/p^{n}\Gamma)$ is an
elementary abelian $p$-group of rank $(n-1)(p^{n}-1)+\tau(p)$. Since
$pJ\subseteq I$, for any $k\geq n+1$, $K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$ maps
onto $K_{2}(\mathbb{Z}[G]/I)$, note that $K_{2}(\mathbb{Z}[G]/(p^{k}))$ maps
onto $K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$, it suffices to prove both
$K_{2}(\mathbb{Z}[G]/(p^{k}))$ and $K_{2}(\mathbb{Z}[G]/I)$ have the same
$p$-rank, this follows from Theorem 2.6 and Lemma 3.3, which completes our
proof. ∎

######  Remark 3.5.

From the proof, we conclude that for any $k\geq n+1$,
$K_{2}(\mathbb{Z}[G]/(p^{k}))$ is isomorphic to
$K_{2}(\mathbb{Z}[G]/p^{k}\Gamma)$.

##  4 Applications.

In [5], the following exact sequence associated to the pair
$(\mathbb{Z}[G],p\mathbb{Z}[G])$:

$K_{2}(\mathbb{Z}[G])\rightarrow K_{2}(\mathbb{F}_{p}[G])\rightarrow
SK_{1}(\mathbb{Z}[G],p\mathbb{Z}[G])\rightarrow
SK_{1}(\mathbb{Z}[G])\rightarrow 1,$

was used to give lower bounds of the $p$-rank of $K_{2}(\mathbb{Z}[G])$ and
$Wh_{2}(G)$, where $G$ is an elementary abelian $p$-group of rank $n$.
However, we find the estimate is inaccurate when $p$ is an odd prime since the
explicit structure of $SK_{1}(\mathbb{Z}[G],p\mathbb{Z}[G])$ is unknown. In
this section, we will give the explicit structure of
$SK_{1}(\mathbb{Z}[G],(p^{k}))$ for $k\geq 1$, and new estimates of the lower
bounds of the $p$-rank of $K_{2}(\mathbb{Z}[G])$ and $Wh_{2}(G)$.

Let $G$ be a finite abelian $p$-group, $\Gamma$ the maximal $\mathbb{Z}$-order
in $\mathbb{Q}[G]$, and $\chi:G\rightarrow\mathbb{C}^{*}$ an irreducible
character of $G$. $\chi(G)$ is a finite cyclic group of roots of unity whose
order dividing $|G|$, and $\chi$ induces a surjective homomorphism from
$\mathbb{Q}[G]$ to $\mathbb{Q}[\chi]$, where $\mathbb{Q}[\chi]$ denotes the
cyclotomic extension of $\mathbb{Q}$ generated by $\chi(G)$. Recall that a
cluster of characters $S$ for $G$ is any set of irreducible characters which
contains exactly one character dividing each irreducible
$\mathbb{Q}[G]$-module (see [2, Definition 2.1]), and there is a natural
isomorphism $\mathbb{Q}[G]\cong\prod_{\chi\in S}\mathbb{Q}[\chi],$
furthermore, by [2, Proposition 2.2], $\Gamma\cong\prod_{\chi\in
S}\mathbb{Z}[\chi]$, and $|G|\Gamma\subseteq\mathbb{Z}[G]\subseteq\Gamma$. An
imaginary cluster of characters for $G$ is defined to be $S_{0}=\\{\chi\in
S|\mathbb{Z}[\chi]\neq\mathbb{Z}\\}$, obviously,

$S_{0}=\left\\{\begin{array}[]{ccl}\\{\chi\in
S\mid|\mbox{Im}(\chi)|>2\\}&&{\mbox{if $p=2$},}\\\ S\backslash\\{\mbox{trivial
character}\\}&&{\mbox{if $p\neq 2$}.}\\\ \end{array}\right.$

######  Lemma 4.1.

[7, Lemma 13.37] If $R$ is a commutative ring with ideals
$J^{{}^{\prime}}\subseteq J$ of finite index, then inclusion on $SL$ induces a
surjective homomorphism $SK_{1}(R,J^{{}^{\prime}})\rightarrow SK_{1}(R,J)$.

We first give the structure of the inverse limit of
$SK_{1}(\mathbb{Z}[G],(p^{k}))$:

######  Proposition 4.2.

Let $G$ be a finite abelian $p$-group, $S_{0}$ an imaginary cluster of
characters for $G$. Then

$\lim\limits_{\overset{\longleftarrow}{k}}SK_{1}(\mathbb{Z}[G],(p^{k}))\cong\prod_{\chi\in
S_{0}}\mbox{Im}(\chi).$

######  Proof.

Set $|G|=p^{k_{0}}$, then for any $k\geq k_{0}$, $p^{k}\Gamma\subseteq
p^{k-k_{0}}\mathbb{Z}[G]\subseteq p^{k-k_{0}}\Gamma$, therefore

$\displaystyle\lim\limits_{\overset{\longleftarrow}{k}}SK_{1}(\mathbb{Z}[G],(p^{k}))$
$\displaystyle=\lim\limits_{\overset{\longleftarrow}{k\geq
k_{0}}}SK_{1}(\mathbb{Z}[G],(p^{k}))\cong\lim\limits_{\overset{\longleftarrow}{k\geq
k_{0}}}SK_{1}(\mathbb{Z}[G],p^{k}\Gamma)$ (by Lemma 4.1)
$\displaystyle\cong\lim\limits_{\overset{\longleftarrow}{k\geq
k_{0}}}SK_{1}(\Gamma,p^{k}\Gamma)$ (by [2, Theoerm 1.3(f)])
$\displaystyle\cong\lim\limits_{\overset{\longleftarrow}{k\geq
k_{0}}}\prod_{\chi\in S}SK_{1}(\mathbb{Z}[\chi],p^{k}\mathbb{Z}[\chi])$
$\displaystyle\cong\lim\limits_{\overset{\longleftarrow}{k\geq
k_{0}}}\prod_{\chi\in
S}SK_{1}(\mathbb{Z}[\chi],p^{k}\mathbb{Z}[\chi])\cong\prod_{\chi\in
S_{0}}\mbox{Im}(\chi).$ (by Corollary 4.3 in [3])

∎

######  Theorem 4.3.

Let $p$ be an odd prime, $G$ an elementary abelian $p$-group of rank $n$. Then
$SK_{1}(\mathbb{Z}[G],p\mathbb{Z}[G])$ is an elementary abelian $p$-group of
rank $\dfrac{p^{n}-1}{p-1}-n$, for any $k\geq 2$,
$SK_{1}(\mathbb{Z}[G],p^{k}\mathbb{Z}[G])$ is an elementary abelian $p$-group
of rank $\dfrac{p^{n}-1}{p-1}$.

######  Proof.

Since $SK_{1}$ of finite rings are trivial, we have the following natural
commutative diagram with exact rows of $K$-groups:

$\setcounter{MaxMatrixCols}{11}\begin{CD}K_{2}(\mathbb{Z}[G])@>{}>{}>K_{2}(\mathbb{Z}[G]/(p^{k+1}))@>{}>{}>SK_{1}(\mathbb{Z}[G],(p^{k+1}))@>{}>{}>SK_{1}(\mathbb{Z}[G])@>{}>{}>1\\\
\Big{\|}@V{}V{\cong}V@V{}V{}V\Big{\|}\Big{\|}\\\
K_{2}(\mathbb{Z}[G])@>{}>{}>K_{2}(\mathbb{Z}[G]/(p^{k}))@>{}>{}>SK_{1}(\mathbb{Z}[G],(p^{k}))@>{}>{}>SK_{1}(\mathbb{Z}[G])@>{}>{}>1\\\
\end{CD}$

By Theorem 2.6, $K_{2}(\mathbb{Z}[G]/(p^{k}))$ is stable for $k\geq 2$, then a
Snake Lemma argument shows that $SK_{1}(\mathbb{Z}[G]/(p^{k}))$ is also stable
for $k\geq 2$, and hence coincides with its inverse limit, thus
$SK_{1}(\mathbb{Z}[G],(p^{k}))$ is an elementary abelian $p$-group of rank
$\dfrac{p^{n}-1}{p-1}$ by Lemma 4.2. To complete the proof, it suffices to
compute $SK_{1}(\mathbb{Z}[G],(p))$. Write $G=\prod_{i=1}^{n}G_{i}$, where
$G_{i}$ denotes a cyclic group of order $p$, and let $\Gamma_{i}$ be the
maximal $\mathbb{Z}$-order of $\mathbb{Z}[G_{i}]$, by the proof of Proposition
4.2, we have

$SK_{1}(\mathbb{Z}[G_{i}],p^{k}\Gamma_{i})\cong
SK_{1}(\Gamma_{i},p^{k}\Gamma_{i})\cong
SK_{1}(\mathbb{Z},p^{k}\mathbb{Z})\oplus
SK_{1}(\mathbb{Z}[\zeta_{p}],p^{k}\mathbb{Z}[\zeta_{p}])=C_{p}.$

Note that there is a natural commutative diagram of $K$-groups:

$\begin{CD}K_{2}(\mathbb{Z}[G]/(p^{k}))@>{\partial}>{}>SK_{1}(\mathbb{Z}[G],(p^{k}))\\\
@V{p_{i*}}V{}V@V{p_{i*}}V{}V\\\
K_{2}(\mathbb{Z}[G_{i}]/(p^{k}))@>{\partial}>{}>SK_{1}(\mathbb{Z}[G_{i}],(p^{k}))\end{CD}$
(4.1)

in which $\partial$ denotes the connecting homomorphism in the algebraic
$K$-theory sequence, $p_{i*}$ denotes the map induced by the projection
$p_{i}:G\rightarrow G_{i}$. Write
$L=\mbox{Ker}(K_{2}(\mathbb{Z}[G]/(p^{k}))\rightarrow
K_{2}(\mathbb{F}_{p}[G]))$, then from the proof of Theorem 2.6, $L$ is an
elementary abelian $p$-group of rank $n$, and

$L\cong\prod_{i=1}^{n}K_{2}(\mathbb{Z}[G_{i}]/(p^{k})),$

since $SK_{1}(\mathbb{Z}[G_{i}])$ is trivial, applying the algebraic
$K$-theory sequence associated to the pair $(\mathbb{Z}[G_{i}],(p^{k}))$,
$K_{2}(\mathbb{Z}[G_{i}]/(p^{k}))$ maps onto
$SK_{1}(\mathbb{Z}[G_{i}],(p^{k}))$, while by Lemma 4.1,
$SK_{1}(\mathbb{Z}[G_{i}],(p^{k}))$ maps onto
$SK_{1}(\mathbb{Z}[G_{i}],p^{k}\Gamma_{i})$, which is a cyclic group of order
$p$, while by Lemma 2.4, $K_{2}(\mathbb{Z}[G_{i}]/(p^{k}))$ is a cyclic group
of order $p$, hence

$K_{2}(\mathbb{Z}[G_{i}]/(p^{k}))\cong
SK_{1}(\mathbb{Z}[G_{i}],(p^{k}))=C_{p}.$

Since the square (4.1) commutes and $SK_{1}(\mathbb{Z}[G],(p^{k}))$ has
exponent $p$, $SK_{1}(\mathbb{Z}[G_{i}],(p^{k}))$ is isomorphic to a direct
summand of $SK_{1}(\mathbb{Z}[G],(p^{k}))$. Let $L^{{}^{\prime}}$ be the image
of $L$ in $SK_{1}(\mathbb{Z}[G],(p^{k}))$, then $L^{{}^{\prime}}$ is an
elementary abelian $p$-group of rank $n$, and

$L^{{}^{\prime}}\cong\prod_{i=1}^{n}SK_{1}(\mathbb{Z}[G_{i}],(p^{k})),$

therefore, for any $k\geq 2$, we have another commutative diagram with exact
rows:

$\setcounter{MaxMatrixCols}{11}\begin{CD}K_{2}(\mathbb{Z}[G])@>{}>{}>K_{2}(\mathbb{Z}[G]/(p^{k}))/L@>{}>{}>SK_{1}(\mathbb{Z}[G],(p^{k}))/L^{{}^{\prime}}@>{}>{}>SK_{1}(\mathbb{Z}[G])@>{}>{}>1\\\
\Big{\|}@V{}V{\cong}V@V{}V{}V\Big{\|}\Big{\|}\\\
K_{2}(\mathbb{Z}[G])@>{}>{}>K_{2}(\mathbb{Z}[G]/(p))@>{}>{}>SK_{1}(\mathbb{Z}[G],(p))@>{}>{}>SK_{1}(\mathbb{Z}[G])@>{}>{}>1\\\
\end{CD}$

by definition, the second vertical arrow is an isomorphism, then by Snake
Lemma, the third vertical arrow is also an isomorphism, hence
$SK_{1}(\mathbb{Z}[G],(p))$ is an elementary abelian $p$-group of rank
$\dfrac{p^{n}-1}{p-1}-n$. ∎

######  Remark 4.4.

For $k\geq 2$, from the proof of Proposition 4.3, we have
$SK_{1}(\mathbb{Z}[G],(p^{k}))\cong SK_{1}(\Gamma,(p^{k}))$, while according
to Theorem A.2 and Theorem A.3 in the appendix of [2], we have
$SK_{1}(\Gamma,(p^{k}))\cong K_{2}(\Gamma/(p^{k}))$, hence by [1, Proposition
2.5], the image of $K_{2}(\mathbb{Z}[G]/(p^{k}))$ in
$SK_{1}(\mathbb{Z}[G],(p^{k}))$ is an elementary abelian $p$-group of rank
$\dbinom{p+n-1}{p}$, for $k\geq 1$, the image has $p$-rank
$\dbinom{p+n-1}{p}-n$. Hence we rediscover the classical result that
$SK_{1}(\mathbb{Z}[G])$ is an elementary abelian $p$-group of rank
$\dfrac{p^{n}-1}{p-1}-\dbinom{p+n-1}{p}$.

With minor modifications of the proofs of Theorem 8 and Corollary 9 in [5],
one can give more precise estimations of the lower bounds of the $p$-rank of
$K_{2}(\mathbb{Z}[G])$ and $Wh_{2}(\mathbb{Z}[G])$.

######  Corollary 4.5.

Let $p$ be an odd prime, $G$ an elementary abelian $p$-group of rank $n$. Then
$K_{2}(\mathbb{Z}[G])$ has $p$-rank at least

$1+(n-1)p^{n}-\dbinom{p+n-1}{p},$

and $Wh_{2}(G)$ has $p$-rank at least

$1+(n-1)p^{n}-\dbinom{p+n-1}{p}-\dfrac{n(n-1)}{2}.$

######  Remark 4.6.

It follows that the lower bound of the order of $K_{2}(\mathbb{Z}[G])$
coincides with [8, Theorem 5.3], since the order is estimated by the order of
the image of $K_{2}(\mathbb{Z}[G])$ in $K_{2}(\mathbb{Z}[G]/(p^{k}))$.
Besides, using exact sequence (1.3) and the explicit structure of
$K_{2}(\mathbb{Z}[G]/p^{n+1}\Gamma)$ (see Theorem 3.4), one can rediscover our
previous result (see [12, Theorem 2.7]), which is a better estimation, the
differences occur when $p$ is an irregular prime.

## Acknowledgements

We thank the referees for their time and comments.

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