# Wideband and Entropy-Aware Deep Soft Bit Quantization ††thanks: This work
was supported by NSF IFML 2019844, ONR grant N00014-19-1-2590, a UT Austin
Machine Learning Lab Research Award, and an AWS Machine Learning Research
Award.

Marius Arvinte  Electrical and Computer Engineering
University of Texas at Austin
Austin, TX, USA
<EMAIL_ADDRESS>Jonathan I. Tamir  Electrical and Computer Engineering
University of Texas at Austin
Austin, TX, USA
<EMAIL_ADDRESS>
###### Abstract

Deep learning has been recently applied to physical layer processing in
digital communication systems in order to improve end-to-end performance. In
this work, we introduce a novel deep learning solution for soft bit
quantization across wideband channels. Our method is trained end-to-end with
quantization- and entropy-aware augmentations to the loss function and is used
at inference in conjunction with source coding to achieve near-optimal
compression gains over wideband channels. To efficiently train our method, we
prove and verify that a fixed feature space quantization scheme is sufficient
for efficient learning. When tested on channel distributions never seen during
training, the proposed method achieves a compression gain of up to $10\%$ in
the high SNR regime versus previous state-of-the-art methods. To encourage
reproducible research, our implementation is publicly available at
https://github.com/utcsilab/wideband-llr-deep.

###### Index Terms:

Deep Learning, Soft Bits, Quantization

##  I Introduction

Soft bit quantization [1, 2] is an important task in integrated, low-power
digital communication platforms where memory is an expensive asset [3]. A
critical application area for quantizing estimated soft bits is given by
hybrid automatic repeat request (HARQ) schemes in, e.g., 5G networks [4],
where information from a failed transmission is stored in order to boost the
performance via soft combining methods [5]. Given that a cellular base station
may communicate with hundreds of users simultaneously, storing soft bits from
failed packets requires efficient and low-distortion quantization methods to
avoid memory bottlenecks on the platform. Another application area where a
flexible trade-off between compression rate and reconstruction distortion is
desirable is given by compress-and-forward relaying schemes [6, 7], where
estimated soft bits are forwarded to a receiver and compressed in order to
lower relay channel resource utilization.

The recent success of deep learning applied to compression problems [8]
motivates us to develop deep soft bit quantization methods. A major challenge
here is given by the fact that any hard quantization operator has zero
gradient almost everywhere, and thus cannot be used in conjunction with modern
optimization algorithms. To this end, various types of practical
approximations and solutions have been developed [9, 8, 10] to make deep
neural networks quantization-aware, and in this work, we use the pass-through
gradient estimation approach [9] due to its simplicity and ease of
implementation.

In this paper, we introduce a data-driven approach for soft bit quantization
over wideband channels. Our scheme consists of a lightweight, properly
initialized deep autoencoder network, and is trained for soft bit
reconstruction in random channels using a differentiable approximation to
quantization, as well as a continuous approximation to the entropy of a
discrete source. During inference, lossless source coding is performed over
the latent representations of soft bits from a wideband channel transmission
to maximize compression gains. Experimental results over simulated EPA [11]
channel realizations demonstrate state-of-the-art performance of the proposed
approach, as well as a controllable trade-off between compression rate and
distortion.

###  I-A Related Work

Prior work on soft bit quantization generally belongs in one of two
categories: classical methods [2, 1, 12] develop near-optimal scalar
quantization methods directly in the log or hyperbolic tangent domain. In
particular, the method in [12] introduces an optimal scalar quantization
method for soft bits that supports a data-driven formulation and learns a
codebook that maximizes the mutual information between the original and
reconstructed soft bits. While this is optimal for scalar (per soft bit
position) quantization, it does not take advantage of redundancy in soft bits
derived from the same or correlated channels.

Recently, the work in [13] introduces an architecture for deep soft bit
quantization, building on the observation that the soft bits corresponding to
a high-order modulation scheme transmitted over a single channel are
correlated, and can always be represented exactly with three values,
regardless of the modulation order. This motivates a deep learning approach in
which an autoencoder is trained to compress the soft bits, which are further
numerically quantized at inference time. Our work builds directly upon [13],
with the following important distinctions: (i) our approach is entropy- and
quantization-aware during training, (ii) during inference, we apply source
coding to compress soft bits, and (iii) provide a tunable, continuous trade-
off between compression rate and distortion.

###  I-B Contributions

Summarized, our contributions are the following:

  1. 1.

We introduce a deep soft bit quantization architecture that is quantization-
aware through a differentiable approximation used in the backwards pass and
entropy-aware through a soft entropy that is annealed over the course of
training.

  2. 2.

We derive the exact variance for the latent representation of the soft bits in
the asymptotically large signal-to-noise ratio (SNR) regime, at
initialization, in a deep neural network with one hidden layer and ReLU
activation. This is used for the one-time design of a fixed quantization
codebook and helps stabilize learning.

  3. 3.

We experimentally demonstrate state-of-the-art quantization performance and
rate-distortion trade-off in realistic wideband channel models, when compared
against classical and deep learning baselines, on a channel distribution that
is completely unseen during training.

##  II System Model

Figure 1: Diagram of the proposed wideband quantization approach. Blocks in
yellow are active during both training and inference (real-world deployment).
The pass-through approximation and entropy estimation are only performed for
training purposes. During inference, source coding is applied to the entire
wideband quantized latent matrix $\mathbf{z}_{Q}$, and the binary codeword is
stored for future use at a potentially different location that has access to
the decoder $g$.

We consider a communications model that transmits a number of $N$ parallel
data channels to a single user, such as sub-carriers in an orthogonal
frequency division multiplexing (OFDM) scenario. Assuming no cross-channel
interference, the signal received on the $i$-th channel is given by the linear
model [14, Eq. 3.1]:

$y_{i}=h_{i}x_{i}+n_{i},$ (1)

where $h_{i}$ is the channel gain, $x_{i}$ is the transmitted symbol and
$n_{i}$ is the noise corresponding to the $i$-th subcarrier, with all
variables being complex-valued. We assume that the noise is drawn from a
complex, circular Gaussian distribution with zero mean and standard deviation
of $\sigma_{n}$. We assume that the symbols $x_{i}$ are obtained by mapping a
set of $K$ bits $\\{b_{k}\\}_{k=1}^{K}$ to a complex-valued constellation
symbol, which is the common practice of digital modulation. Given ideal
channel knowledge $h_{i}$ and known noise statistics $\sigma_{n}$, and
assuming that transmitted bits are sampled i.i.d. with equal probabilities,
the maximum likelihood (ML) estimate of the log-likelihood ratio for the
$k$-th bit transmitted on the $i$-th channel is given by:

$L_{i,k}=\log\frac{P(y_{i}|b_{i,k}=1)}{P(y_{i}|b_{i,k}=0)}=\log\frac{{\sum\limits_{s\in\\{s|b_{k}=1\\}}}\exp{-\frac{|y_{i}-h_{i}s|^{2}}{\sigma_{n}^{2}}}}{{\sum\limits_{s\in\\{s|b_{k}=0\\}}}\exp{-\frac{|y_{i}-h_{i}s|^{2}}{\sigma_{n}^{2}}}}.$
(2)

The soft bits are defined as $\Lambda_{i,k}=\tanh{\frac{L_{i,k}}{2}}$ and
grouped in the wideband soft bit matrix $\mathbf{\Lambda}$. The goal of
wideband quantization is to design the triplet of functions
$(f,g,\mathcal{Q})$, where the encoder $f$ maps the floating point input
$\mathbf{\Lambda}$ to a latent representation, $\mathcal{Q}$ maps this
representation to a finite bit string, and the decoder $g$ recovers
$\mathbf{\Lambda}$ with minimal distortions. Note that this scheme maps the
entire soft bit matrix to a single binary codeword, and is thus a form of
vector quantization. The compression rate and the distortion of the
reconstruction are denoted by the functions:

$\mathcal{R}=H(f(\mathbf{\Lambda}))\ \text{and}\
\mathcal{D}=\frac{1}{NK}\sum_{i,k}\frac{|\Lambda_{i,k}-\tilde{\Lambda}_{i,k}|^{2}}{|\Lambda_{i,k}|+\epsilon},$
(3)

respectively, where $H$ represents the entropy of a discrete source expressed
in bits, and $\tilde{\mathbf{\Lambda}}=g(f(\mathbf{\Lambda}))$ is the
recovered soft bit matrix and $\epsilon$ is a small numerical constant used to
prevent overflow. We use a sample-weighted version of the mean squared error,
which is a pseudo-metric since it is not symmetric and does not satisfy the
triangle inequality property. This choice is taken from [13], since minimizing
this metric places more importance on uncertain soft bits and benefits the
decoding of error-correcting codes [15].

##  III Proposed Method

Fig. 1 shows an overview of the proposed approach for wideband, entropy-aware
soft bit quantization. During training, the model uses the wideband soft bit
matrix $\mathbf{\Lambda}$ and a soft entropy estimate of the quantized
representation to optimize the weights of $f$ and $g$, which are deep neural
networks. During inference, lossless source coding is applied to the quantized
representation $\mathbf{z}_{Q}$ to reduce storage costs to near-entropy
levels. The resulting binary string is stored until decoding is required,
e.g., in hybrid ARQ or relay scenarios. In the following, we give a
description of each of the involved components.

###  III-A Encoder

The encoder $f$ is a function that maps an input matrix $\mathbf{\Lambda}$ to
a latent matrix $\mathbf{z}$ by applying the same functional backbone in a
row-wise manner and stacking the representations in a matrix:

$\mathbf{z}=\textrm{stack}_{i}(f(\mathbf{\Lambda}_{i})).$ (4)

We design the backbone of $f$ as a fully-connected, feed-forward network with
ReLU activation in the hidden layers and $\tanh{x}$ as the output activation.
The input size is a vector of size $K$, the hidden layers are all of size
$4K$, while the output is of fixed size equal to three. This corresponds to
the universal latent dimension of a soft bit vector with arbitrary $K$, as
introduced in [13]. That is, without quantization, such a compressive
representation (from $K$ soft bits to three latent variables) is guaranteed to
exist and can be represented and learned by a deep neural network with a
sufficient modeling capacity.

###  III-B Latent Quantization

This block applies a discrete quantization operator $\mathcal{Q}$ to each
component of the latent representation $\mathbf{z}$ in the forward pass of the
network. During training, since the quantization operator has zero gradient
almost everywhere, we use a pass-through approximation [9] to obtain a
differentiable function for the backward pass. That is, the forward and
backward pass signals are, respectively:

$\begin{split}z_{Q,i,\textrm{fw}}&=\mathcal{Q}(z_{i}),\\\
z_{Q,i,\textrm{bw}}&=\texttt{sg}[\mathcal{Q}(z_{i})-z_{i}]+z_{i},\end{split}$
(5)

where $\mathbf{z}_{Q,\textrm{fw}}$ and $\mathbf{z}_{Q,\textrm{bw}}$ are the
forward and backward pass latent signals, respectively, and sg is the stop-
gradient operator, which prevents gradient from flowing in the backward pass.
This leads to the gradient of the quantized representation with respect to its
input being $\frac{\partial z_{Q,i,\textrm{bw}}}{\partial z_{i}}=1$ and allows
gradients to propagate to earlier layers.

Importantly, the function $\mathcal{Q}$ is a pre-determined scalar
quantization function that is held fixed throughout learning and inference.
This differentiates us from [13] and [8] and enables efficient learning, given
a careful choice of $\mathcal{Q}$. In the following, we present a theoretical
and empirical analysis of deep neural networks that are used for soft bit
quantization in the high SNR regime and show that a choice for $\mathcal{Q}$
that avoids the issue of codebook collapse [16, 17] – where a portion of the
codebook may never be used during training – can be found at initialization.
We use the two following lemmas in our proof:

###### Lemma 1

Let $\mathbf{X}\in\mathbb{R}^{m\times n}$ be a matrix with i.i.d. Gaussian
elements and let $\mathbf{Y}\in\mathbb{R}^{n}$ be an i.i.d. Rademacher random
variable, independent of $\mathbf{X}$. Then, the elements of
$\mathbf{X}\mathbf{Y}$ are distributed as i.i.d. Gaussian random variables.

###### Proof:

Immediate by the independence of $\mathbf{X}$ and $\mathbf{Y}$. ∎

###### Lemma 2

Let $X$ be a scalar random variable distributed as $\mathcal{N}(0,\sigma)$.
Then, $\mathrm{relu}(X)=\mathrm{max}\\{X,0\\}$ has the following properties:

  * •

$\mathbb{E}[\mathrm{relu}(X)]=\frac{1}{\sqrt{2\pi}}\sigma,$

  * •

$\textrm{Var}(\mathrm{relu}(X))=(\frac{1}{2}-\frac{1}{2\pi})\sigma^{2}.$

###### Proof:

Follows immediately from [18, Page 3] and re-writing $\mathrm{relu}(X)$ as a
mixture of two random variables. ∎

We now state and prove the following the following theorem.

###### Theorem 1

Let $f$ be a one hidden-layer, fully-connected neural network with no biases,
hidden $\mathrm{relu}$ activation, and linear output activation. Let the
length of the input vector $\mathbf{\Lambda}$ be $K$, the hidden size be $4K$,
and the output size be $1$. The weight matrices are
$\mathbf{W}\in\mathbb{R}^{4K\times K}$ and $\mathbf{V}\in\mathbb{R}^{1\times
4K}$, respectively. We make the following assumptions:

  * •

The entries of $\mathbf{\Lambda}$ are drawn i.i.d. from a Rademacher
distribution such that $p(\Lambda_{i}=1)=p(\Lambda_{i}=-1)=0.5$.

  * •

The entries of the hidden layer weight matrix $\mathbf{W}$ are drawn i.i.d.
from a Gaussian distribution with $\mu_{w}=0$ and
$\sigma_{w}=\sqrt{\frac{2}{5K}}$, respectively.

  * •

The entries of the output layer weight vector $\mathbf{V}$ are drawn i.i.d.
from a Gaussian distribution with $\mu_{v}=0$ and
$\sigma_{v}=\sqrt{\frac{2}{4K+1}}$, respectively.

Let $z=\sum_{i}v_{i}\mathrm{relu}(\mathbf{W}\mathbf{\Lambda})_{i}$ be the
output of the network. Then, it satisfies the following properties:

  * •

$\mathbb{E}[z]=0,$

  * •

$\textrm{Var}(z)=\frac{8}{5}\frac{K}{4K+1}.$

###### Proof:

Using Lemma 1 and the first two assumptions, it follows that the pre-
activation values after the first layer are Gaussian distributed. Using Lemma
2 on these activations allows us to characterize the mean and standard
deviation of $\mathrm{relu}(\mathbf{W}\mathbf{\Lambda})_{i}$. Since the
entries $\mathbf{W}\mathbf{\Lambda}$ are i.i.d., it follows that the entries
of $\mathrm{relu}(\mathbf{W}\mathbf{\Lambda})$ are also i.i.d., and also
independent from the weights $\mathbf{V}$, as well as using the second and
third assumptions. Since $\mathbf{V}$ is zero-mean, we obtain that:

$\begin{split}\textrm{Var}(z)&=\textrm{Var}\left(\sum_{i}v_{i}\mathrm{relu}(\mathbf{W}\mathbf{\Lambda})_{i}\right)\\\
&=\sum_{i}\textrm{Var}\left(v_{i}\mathrm{relu}(\mathbf{W}\mathbf{\Lambda})_{i}\right)\\\
&=\sum_{i}\textrm{Var}(v_{i})\textrm{Var}\left(\mathrm{relu}(\mathbf{W}\mathbf{\Lambda})_{i}\right)\
+\\\ &\ \ \
\textrm{Var}(v_{i})\mathbb{E}\left[\mathrm{relu}(\mathbf{W}\mathbf{\Lambda})_{i}\right]^{2}\\\
&=4K\sigma_{v}^{2}(\sqrt{K}\sigma_{w})^{2}[(\frac{1}{2}-\frac{1}{2\pi})+\frac{1}{2\pi}]\\\
&=2K^{2}\sigma_{v}^{2}\sigma_{w}^{2}\\\
&=\frac{8}{5}\frac{K}{4K+1}.\end{split}$ (6)

∎

Figure 2: Empirical verification of Theorem 1 across a varying number of $K$
values, corresponding to different modulation orders. The variables
$\hat{\sigma}_{z}$ represent empirical estimates of the latent standard
deviation at initialization, while the stars mark the theoretical values (only
available for one hidden layer). The variables $\hat{P}_{99.9,z}$ represent
the 99.9 percentile values of the absolute latent variables at initialization.

The first assumption corresponds to operating in the asymptotically large SNR
regime, where soft bits tend toward polarized values in the hyperbolic tangent
domain. We study this regime since it allows exact analysis of the variance
and serves as an upper bound for lower SNR regimes in terms of the latent
space variance, since the distribution of the soft bits there is more biased
toward zero and has a reduced variance.

The last two assumptions concern the deep neural network at initialization and
match the Glorot weight initialization scheme [19]. As motivated in [19], this
initialization is carefully chosen such that the variance of the signal is
reduced as the network gets deeper. To verify Theorem 1 and the empirical
reduction of variance, we plot the estimated standard deviation of
$\mathbf{z}$ and the estimated range of $\mathbf{z}$ in Fig. 2 for a varying
depth of the network and different values of $K$. The architecture follows the
exact assumptions of Theorem 1, and is the basis for the model we use in
practice.

The exact match between the empirical $\hat{\sigma}_{z}$ and the starred
points verifies Theorem 1 for a network with one hidden layer, and these
values are almost invariant to $K$ due to the ratio in (6). Fig. 2 also plots
the $99.9$ percentile values for an increasing number of hidden layers. While
an exact analysis is out of scope here, we find that using the Glorot
initialization leads to a decreasing latent variance as the network gets
deeper, as originally pointed out in [19]. Since the encoder $f$ uses a
$\mathrm{tanh}$ function as activation, this is extremely useful in preventing
latent collapse – the presence of strong modes at $\pm 1$ – and allows the use
of a fixed quantizer $\mathcal{Q}$ throughout the entire training process.

###  III-C Entropy Estimation

Given $\mathbf{z}$, $\mathbf{z}_{Q}$ and a quantization codebook $Q$ with $M$
entries, we estimate the $\tau$-soft entropy [8] as:

$\mathcal{H}(\mathbf{z};\tau)=-\frac{1}{N}\sum_{i=1}^{M}\sum_{j=1}^{N}\phi_{i}(z_{j};\tau)\log{p_{i}},$
(7)

where $q_{i,j}=\phi_{i}(z_{j};\tau)$ represents the soft allocation of $z_{j}$
to the $i$-th entry in the quantization codebook. That is,
$q_{i,j}=\textrm{softmax}_{i}\big{(}-\frac{|z_{j}-Q_{i}|^{2}}{\tau}\big{)}$,
where the softmax is taken across all codebook entries and $\tau$ represents
the inverse temperature of this approximation. The terms $p_{i}$ represent the
empirical probability estimates of $\mathbf{z}_{Q}$ obtained by counting over
$N$ samples. Hence, no gradient flows through the $\log{p_{i}}$ term during
training. An important aspect here is that as $\tau\rightarrow\infty$ and the
sample size $N$ is sufficiently large, we have that
$\mathcal{H(\mathbf{z};\tau)}\rightarrow
H(\mathbf{z}_{Q})=-\sum_{i}p_{i}\log{p_{i}}$, the entropy of the discrete
random variable $\mathbf{z}_{Q}$.

###  III-D Source Coding

Given a quantized representation $\mathbf{z}_{Q}$, the discrete probabilities
$p_{i}$ for all codebook symbols are estimated from feature representations of
a fixed, finite set of training channels, and lossless source coding is
applied during inference for storage or relaying purposes. Our method is
compatible with any source coding scheme. In practice, we use arithmetic
coding [20] due to its near-optimal performance and extremely efficient
publicly available implementation [21]. Since the coding is lossless, there is
no incurred performance loss.

###  III-E Decoder

The decoder $g$ is a deep neural network with an architecture that mirrors
$f$, including the number of layers and the hidden dimension. That is, it maps
an input latent matrix $\mathbf{z}$ to the reconstructed soft bit matrix
$\tilde{\mathbf{\Lambda}}$ by applying the shared layers in a row-wise manner:

$\tilde{\mathbf{\Lambda}}=\textrm{stack}_{i}(g(\mathbf{z_{i}})).$ (8)

Given all components, the model is trained with the end-to-end supervised
loss:

$L(\mathbf{\Lambda},\tilde{\mathbf{\Lambda}};\tau)=\mathcal{D}(\mathbf{\Lambda},\tilde{\mathbf{\Lambda}})+\alpha\mathcal{H}(\mathbf{z};\tau).$
(9)

The first term corresponds to the quantization-aware reconstruction loss that
ensures soft bits are recovered properly after numerical quantization of the
latent representation $\mathbf{z}$. The second term serves as a approximation
for minimizing the entropy of the quantized latent representation
$\mathbf{z}_{Q}$ and to enable further gains with source coding, where
$\alpha$ is a hyper-parameter that directly controls the rate-distortion
trade-off.

##  IV Experimental Results

###  IV-A Architecture and Training

We use deep neural networks for $f$ and $g$, each with four hidden layers,
$\mathrm{relu}$ hidden activations, $\tanh$ activation at the output (for both
$f$ and $g$), and a hidden size of $4K$, where $K$ is the modulation order for
which we train the method – as well as the input size to the network. The
latent dimension is always three and we initialize all layers with the Glorot
scheme [19] to match the conditions of Theorem 1. Complete details about the
architecture are found in our code repository linked in the abstract.

The latent quantizer $\mathcal{Q}$ uniformly covers the interval $[-0.8,0.8]$
using a number of $64$ codebook entries ($6$ bits), and remains fixed
throughout the entire training and inference procedures. The same
$\mathcal{Q}$ is used for all the latent dimensions and the choice of the
interval is a direct consequence of the range of the latent representation
under the $\tanh$ operator, as shown in Fig. 2. It can be seen that the latent
code is bound to this interval, hence no codebook collapse occurs at
initialization.

The data used to train all models comes from transmissions across i.i.d.
Rayleigh fading channels, where $h_{i}\sim\mathcal{N}_{\mathbb{C}}(0,1)$ and
the noise $n_{i}\sim\mathcal{N}_{\mathbb{C}}(0,\sigma_{n})$. Payloads are
generated by randomly sampling bits with equal probabilities and codewords are
obtained by using a low-density parity check (LDPC) code of size $(324,648)$,
for a total of $100000$ training codewords at uniformly spaced SNR values.
Importantly, our method is only trained on soft bits from i.i.d. channels, and
is not trained on a specific wideband channel distribution. We find that a
range of $\alpha$ between $0.001$ and $0.03$ generally covers the entire rate-
distortion curve, and we anneal $\tau$ at epoch $t$ by the schedule
$\tau_{t}=40\times 1.001^{t}$.

A single network is trained across the entire SNR range, and takes about three
hours for $2000$ epochs (invariant to $K$) on an NVIDIA RTX 2080Ti GPU.
Inference takes less than $1$ ms for an OFDM wideband channel with $108$
subcarriers. Storing the network for $K=6$ takes a total of $82.8$ kB in
floating point precision. During inference, soft bits are quantized and
reconstructed, and belief propagation decoding is performed to obtain a
complete communication chain. We measure end-to-end performance through the
block (codeword) error rate figure.

We train the baseline in [13] by using exactly the same data and backbone
architecture for a fair comparison. We also compare with the optimal scalar
method in [12] by learning a separate quantization codebook for each soft bit
position, at each SNR value. This partially compensates for the extra
learnable parameters that deep learning methods have.

###  IV-B End-to-End Quantization Performance

Fig. 3 shows the performance of all methods in a $K=6$ (64-QAM) modulation
scheme and EPA wideband channel model with $108$ subcarriers allocated per
codeword, at a carrier frequency of $2$ GHz and channel bandwidth of $10$ MHz.
The number in the parentheses indicates the average cost required to store a
single soft bit, where we average this cost over the range of SNR values that
lead to block error rates between $1$ and $0.001$, since this range is of
practical interest. A key takeaway here is that all methods are calibrated to
produce the same end-to-end performance, with minimal deviations from un-
quantized performance. The proposed approach suffers a performance loss of
$0.18$ dB compared to floating point at a target error rate of $0.01$, and has
the same performance as [13], while achieving an average compression gain of
$7\%$. Both deep learning-based methods greatly surpass the scalar quantizer,
with ours having an average compression gain of $31\%$.

Fig. 4 reveals how quantization cost scales with SNR for the different
methods, as well as the near-optimality of arithmetic coding in a wideband
scenario. For scalar quantization methods such as maximum MI [12], the cost
per soft bit decreases with increasing SNR. Asymptotically, this behaviour is
optimal since as $\textrm{SNR}\rightarrow\infty$, then the soft bits become
discrete binary random variables as in Theorem 1, and one bit per soft bit is
the optimal quantization scheme. This trend is opposite for deep learning
methods, since the same model accommodates the entire SNR regime: there, the
average cost per bit increases as the SNR increases, and this phenomenon is
much more pronounced for the baseline in [13]. We find that the proposed
approach helps counteract this sub-optimality, again due to its entropy
objective in the loss function.

Figure 3: Block error rate as a function of SNR for the proposed method,
baselines and floating point (no quantization), for $K=6$ ($64$-QAM)
modulation in EPA channels with a bandwidth of $10$ MHz. The values in
parentheses indicate the average storage cost per soft bit. Figure 4: Soft bit
quantization cost as a function of operating SNR for the proposed method and
the baselines in EPA channels. The red and black solid lines indicate
baselines without source coding, while the dashed lines are with source
coding. The orange curve is the performance of our method, while the blue dots
indicate the estimated entropy of $\mathbf{z}_{Q}$ (lowest possible rate). For
all methods, no specific EPA channel simulations are used to train the models
or calibrate the source coding.

Fig. 4 also plots the performance of the two baselines with and without
(horizontal lines) source coding. We note that, while source coding benefits
both baselines, the proposed approach still improves compression rates in a
broad SNR range due to the entropy-aware nature of (9). In the high SNR
regime, the proposed method achieves compression gains of up to $10\%$
compared to [13], and the source coding is near-optimal, since it achieves the
entropy marked with circles.

###  IV-C Rate-Distortion Trade-Off

Fig. 5 investigates the impact of $\alpha$ during training our method, for
$K=8$ on EPA channels with $81$ subcarriers (since the bit mapping is denser
than $K=6$, fewer channel uses are required to send a packet). The operating
characteristics of the model are close to the ones in Fig. 3, and extended
results can be found in our code repository.

To control the trade-off between rate and distortion, we vary the $\alpha$
parameter in our loss function between the range of $0.01$ and $0.03$ and
train a separate model at each value. From Fig. 5 it can be noticed that, at
lower SNR values the absolute penalty in end-to-end error is larger if we
quantize aggressively, whereas the error increases are much smaller in the
high SNR regime.

Figure 5: Rate-distortion curves obtained by tuning $\alpha$ in the loss
function, for $K=8$ ($256$-QAM) in EPA channels. In this figure, $\alpha$ is
linearly interpolated between $0.01$ (right-most points) and $0.03$ (left-most
points) with a spacing of $0.005$. The y-axis represents the additive block
error rate incurred against a floating point solution. Each of the curves
represents a specific SNR point.

##  V Conclusion

In this paper, we have introduced a deep learning approach for wideband soft
bit quantization. Our formulation included a fixed quantizer and a
quantization- and entropy-aware training objective, as well as the use of
source coding at inference. Our theoretical results proved that a fixed
quantizer is sufficient for efficient training, and the experiments have shown
state-of-the-art quantization performance in a wide SNR range, as well as
flexibility in controlling the rate-distortion trade-off.

The model is compact and inference is efficient, achieving sub-ms latency for
an entire wideband channel. Our model is also not trained on a specific
channel distribution, but instead can operate on arbitrary wideband channels.
While this provides a degree of flexibility, a promising future research
direction is to investigate whether further compression gains can be obtained
by specializing a model for a specific channel distribution and develop
adaptive quantization schemes.

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# Micro Tidal Disruption Events in Active Galactic Nuclei

Y. Yang  Department of Physics, University of Florida, PO Box 118440,
Gainesville, FL 32611-8440, USA  I. Bartos  Department of Physics, University
of Florida, PO Box 118440, Gainesville, FL 32611-8440, USA  G. Fragione
Department of Physics & Astronomy, Northwestern University, Evanston, IL
60208, USA  Center for Interdisciplinary Exploration & Research in
Astrophysics (CIERA), Northwestern University, Evanston, IL 60208, USA  Z.
Haiman  Department of Astronomy, Columbia University, 550 W. 120th St., New
York, NY, 10027, USA  M. Kowalski  Deutsches Elektronen Synchrotron DESY,
Platanenallee 6, 15738 Zeuthen, Germany  Institut fur Physik, Humboldt-
Universit at zu Berlin, D-12489 Berlin, Germany  S. Márka  Columbia
Astrophysics Laboratory, Columbia University in the City of New York, New
York, NY 10027, USA  R. Perna  Department of Physics and Astronomy, Stony
Brook University, Stony Brook, NY 11794-3800, USA  Center for Computational
Astrophysics, Flatiron Institute, New York, NY 10010, USA  H. Tagawa
Astronomical Institute, Graduate School of Science, Tohoku University, Aoba,
Sendai 980-8578, Japan

###### Abstract

Active galactic nuclei (AGNs) can act as black hole assembly lines, funneling
some of the stellar-mass black holes from the vicinity of the galactic center
into the inner plane of the AGN disk where the black holes can merge through
dynamical friction and gravitational wave emission. Here, we show that stars
near the galactic center are also brought into the AGN disk, where they can be
tidally disrupted by the stellar-mass black holes in the disk. Such micro-
tidal disruption events (micro-TDEs) could be useful probe of stellar
interaction with the AGN disk. We find that micro-TDEs in AGNs occur at a rate
of $\sim 170$ Gpc-3yr-1. Their cleanest observational probe may be the
detection of tidal disruption in AGNs by heavy supermassive black holes
($M_{\bullet}\gtrsim 10^{8}$ M⊙) so that cannot tidally disrupt solar-type
stars. We discuss two such TDE candidates observed to date (ASASSN-15lh and
ZTF19aailpwl).

††thanks<EMAIL_ADDRESS>
##  1 Introduction

A few percent of galaxies host a central, compact, highly luminous region
called active galactic nucleus (AGN). AGNs are the result of a highly
accreting supermassive black hole (SMBH), in which infalling gas forms an
accretion disk around the SMBH.

In addition to being the source of the high central luminosity, AGN disks can
also impact the dynamics of stellar remnants in the galactic center. Of
particular interest has been the interaction of AGN disks with stellar-mass
black holes (BHs) that have migrated to the galactic center through mass
segregation (Alexander & Hopman, 2009; Keshet et al., 2009; O’Leary et al.,
2009). While orbiting the central SMBH, BHs periodically cross the AGN disk
that gradually aligns their orbit with the disk plane (Bartos et al., 2017).
Once in the disk, BHs migrate inwards due to viscous-tidal interactions with
the disk (McKernan et al., 2012; Bellovary et al., 2016). After these
processes brought two BHs in each others’ vicinity, the dense gas of the AGN
disk facilitates gravitational capture and ultimately the binary merger of the
two BHs through dynamical friction.

BH mergers in AGN disks may be an important gravitational-wave source with BH
properties distinct from those expected from other astrophysical mechanisms
(Bartos et al., 2017; Stone et al., 2017; McKernan et al., 2018; Yang et al.,
2019a, c, 2020a, 2020b; Secunda et al., 2019; Tagawa et al., 2020b, a, 2021a,
2021b; Gayathri et al., 2020; Samsing et al., 2020). Merging BHs or neutron
stars in AGN disks might also produce detectable electromagnetic emission,
opening up another way to study AGN-assisted mergers (McKernan et al., 2019;
Kimura et al., 2021; Perna et al., 2021a, b; Zhu et al., 2021a, b).

Stellar orbits are also affected by AGN disks as disk crossings dissipate
orbital energy (Miralda-Escudé & Gould, 2000), which will be particularly
significant for stars on highly eccentric orbits. Some of these stars will
gradually get closer to the central SMBH until they are tidally disrupted by
it. Nonetheless such interactions with the AGN disk may have limited effect on
the overall tidal disruption rate in galaxies with AGNs (MacLeod & Lin 2020,
but see Tadhunter et al. 2017).

Here we examine the orbital alignment of stars with the AGN disk, and its
consequences. Similarly to BHs and neutron stars, some of the stars in
galactic centers will align their orbits with the AGN disk plane. Once in the
plane, stars can form binaries with BHs, leading to their eventual tidal
disruption within the AGN disk. We discuss the rate density of such tidal
disruption events around BHs (hereafter micro-TDEs; Perets et al. 2016), their
expected observational signatures and possible observational indications of
their existence.

##  2 Orbital evolution around AGN disks

Upon crossing the AGN disk, the velocity of stars will change due to the
accretion of matter from the disk. We simulated the orbital evolution of stars
accounting for this velocity change following Yang et al. (2019b). The mass of
infalling gas upon each crossing is given by

$\Delta m_{\rm gas}=v_{\rm rel}t_{\rm cross}R_{\rm cap}^{2}\pi\Sigma/(2H)\,,$
(1)

where $R_{*}$ and $M_{*}$ are the stellar radius and mass, $R_{\rm
cap}=\max\\{R_{*},R_{\rm BHL}\\}$, $R_{\rm BHL}\equiv 2GM_{*}/(v^{2}_{\rm
rel}+c_{s}^{2})$ is the stars’ Bondi-Hoyle-Lyttleton radius, $\Sigma$ and $H$
are the surface density and scale height of the AGN disk respectively, $t_{\rm
cross}\equiv 2H/v_{\rm z}$ is the crossing time, $v_{\rm z}$ is the $z$
component of the stars’ velocity. We found mass loss by a main-sequence star
due to ram pressure exerted by the disk gas to be negligible (Miralda-Escudé &
Kollmeier, 2005).

We considered a geometrically thin, optically thick, radiatively efficient,
steady-state (Shakura & Sunyaev, 1973) but self-gravitating (Sirko & Goodman,
2003) accretion disk. We adopted viscosity parameter $\alpha=0.1$, accretion
rate $\dot{M}_{\bullet}=0.1\dot{M}_{\rm Edd}$ and radiation efficiency
$\epsilon=0.1$. We used a fiducial SMBH mass of $10^{6}$ M⊙.

We computed the changes in the velocity and angular momentum of the star upon
each crossing based on momentum conservation (Yang et al., 2019b):

$\displaystyle\bf{\Delta{v_{*}}}$
$\displaystyle=-\lambda(\bf{v_{*}}-\bf{v_{gas}})=-\lambda\bf{v_{\rm vel}}$ (2)
$\displaystyle\bf{\Delta J}$ $\displaystyle=-\lambda\bf{r}\times\bf{v_{\rm
rel}}\,,$ (3)

where $\bf{v_{\rm rel}}$ is the relative velocity between the star and the
gas, and $\lambda\equiv\Delta M_{\rm gas}/M_{*}$ is a dimensionless factor. We
updated the parameters of the star’s orbit after each crossing according to
the change of velocity and obtained the orbital evolution iteratively.

We assumed that the stars follow a mass-radius relation $R_{*}=1.06(M_{*}/{\rm
M}_{\odot})^{0.945}$ for $M_{*}<1.66M_{\odot}$ and $1.33(M_{*}/{\rm
M}_{\odot})^{0.555}$ for $M_{*}>1.66M_{\odot}$ (Demircan & Kahraman, 1991). We
assumed that the stellar mass follows a Kroupa initial mass function (Kroupa,
2001). We took into account the mass segregation in the spatial distributions
of main sequence stars, adopting $dN/da\propto a^{-3/2-0.5M_{*}/M_{\rm max}}$,
where $a$ is the semi-major axes of the star’s orbit around the SMBH (Keshet
et al., 2009; Alexander & Hopman, 2009; Gondán et al., 2018). We adopted a
maximum stellar mass of $M_{\rm max}=50$ M⊙, which accounts for the fact that
the lifetime of the most massive stars is too short for them to participate in
mass segregation. The total stellar mass within the gravitational influence
radius of the SMBH was taken to be equal to the mass of the SMBH (Miralda-
Escudé & Gould, 2000).

We similarly computed the orbital alignment of BHs and neutron stars with the
AGN disk. For simplicity we adopted the Salpeter initial mass function
$dN/dM\propto M^{-2.35}$ for BHs within the range of $5-50$ M⊙ and a normal
initial mass function $M/M_{\odot}\sim\textit{N}(1.49,0.19)$ for neutron stars
(Özel & Freire, 2016). We assumed that the total mass of the BH population is
$1.6\%$ of the stellar mass in galactic centers and the number of neutron
stars is ten times the number of BHs (Gondán et al., 2018). We additionally
took into account mass segregation in the three-dimensional spatial
distributions of BHs and neutron stars following Gondán et al. (2018).
Otherwise orbital alignment was computed similarly to stars.

##  3 Binary formation, evolution and tidal disruption

Once some of the stars, BHs and neutron stars align their orbits with the AGN
disk, they begin migrating inward analogously to planetary migration in
protostellar disks (e.g., Tagawa et al. 2020a). Both orbital alignment and
migration increase the number density of each type of object in the inner AGN
disk, leading to efficient binary formation through gravitational capture.
Dynamical friction within the gas further facilitates the formation of
binaries and their consecutive hardening.

We considered a binary consisting of a main sequence star and a BH. The binary
typically forms at small separations of only $\sim 10$ times the star’s tidal
radius $R_{\rm t}=R_{*}(M_{\rm bh}/M_{*})^{1/3}$, and could be eccentric at
(or soon after) its birth, either due to its formation (c.f. Funato et al.
2004) and/or subsequently driven to be eccentric by a circumbinary disk (Zrake
et al., 2021; D’Orazio & Duffell, 2021). This could justify an impulsive
disruption ”event” at the pericenter of an ellipse, rather than a slow
circular inspiral and gradual tidal ”peeling”.

Once the binary’s pericenter distance $R_{p}$ approaches $R_{\rm t}$, the star
is tidally disrupted. TDEs around SMBHs typically occur at parabolic
($e\approx 1$) stellar orbits, leading to a tidal tail that will affect long-
term accretion. In addition, partial disruption has a substantially larger
cross-section in the case of TDEs, and should be more common (Stone et al.,
2020). The lower eccentricity of micro-TDEs in AGNs will result in most of the
stellar material remaining in the vicinity of the BH, leading to a more
massive and compact accretion disk.

To qualitatively understand the above tidal disruption process, we performed a
simulation of a micro-TDE using smoothed-particle hydrodynamics (SPH; Springel
2010). We considered a star with mass $M_{*}=1$ M⊙ and a BH with mass $M_{\rm
bh}=10$ M⊙. The star was initially located at $5R_{\odot}$ distance from the
BH. For simplicity, we assumed that the BH-star binary is on a circular orbit.

We adopted a polytrope model to compute the structure of the main sequence
star, assuming that pressure depends on the density via $P=k\rho^{1+1/n}$,
where constant $k$ is chosen to produce a Sun-like star and $n$ is set to be
3, which can describe the structure of a main sequence star. We considered the
self gravity and adopted artificial viscosity in our simulation (Springel,
2010).

Figure 1: Smoothed-particle hydrodynamics simulation of micro-TDE of a binary
system consisting of a 10$M_{\odot}$ BH and a 1$M_{\odot}$ main sequence star,
the initially circular orbit with 5$R_{\odot}$ separation. The period of the
initial orbit is about 0.4 days. We show the normalized surface density of the
debris at different times and give the normalization factor in each subplot.

The outcome of our simulation is shown in Fig 1. We see that tidal instability
rapidly grows and the star is tidally disrupted within a day. Most of the
stellar matter remains gravitationally bound to the BH.

The above simulation does not account for the presence of the mini-disk around
the BH accumulated from the AGN disk prior to stellar disruption. This can be
quantitatively important but we do not expect it to qualitatively change the
outcome. We estimated that stellar mass loss prior to disruption due to
radiation by the accreting BH will be only $\sim 10\%$ of the star’s mass and
therefore will not meaningfully affect the outcome (Zajaček et al., 2020;
Kimura et al., 2021).

##  4 Radiation features of micro-TDEs

TDE and micro-TDE light curves and spectra are not yet well understood. They
also depend on a large number of variables, such as the masses of the black
hole and the star, the eccentricity and impact parameter of the encounter, the
black hole’s spin, and others.

While in some ways similar, TDE and micro-TDE light curves and spectra can
differ due to multiple effects. First, TDEs by SMBHS occur with highly
eccentric orbits ($e\approx 1$), while AGN-assisted micro-TDEs are not
expected to be highly eccentric. This latter case allows a larger fraction of
the stellar matter to remain in the vicinity of the BH, quickly forming a
nearly circular accretion disk (Kremer et al., 2019). Second, the larger
amount of mass and lower black hole mass in the micro-TDE case results in
highly super-Eddington accretion that can exceed the Eddington rate
$\dot{M}_{\rm Edd}\equiv L_{\rm Edd}/c^{2}=2.6\times 10^{-9}M_{\rm BH}$ yr-1
by up to a factor of $10^{5}$ (Kremer et al., 2019). Here, $L_{\rm Edd}$ is
the Eddington luminosity and $c$ is the speed of light. Super-Eddington
accretion results in a geometrically thick accretion disk. A large fraction of
the disk mass is expected to be blown away as a disk wind. Radiation from the
inner accretion disk will escape with a delay, resulting in an extended peak
in the light curve (Kremer et al., 2019). Further delay in reaching peak
luminosity may be expected given that the AGN will surround micro-TDEs with a
dense medium. The delay, and even whether the micro TDE emission can escape
the AGN disk, may depend on the mass of the AGN disk and the location of the
disruption event within the disk (Perna et al., 2021a). In our fiducial disk
model, micro-TDEs are expected mostly in regions around $10^{-4}-10^{-2}$ pc
from the central SMBH, where the disk is ionized and the opacity to scattering
is very high, significantly altering the observed emission. However, BHs in
the AGN disk are expected to open cavities due to accretion-induced radiation
(Kimura et al., 2021), which results in the reduction of opacity, leaving
emission from the micro-TDE less affected. Finally, the vicinity of the SMBH
for binaries that reach the AGN disk’s migration trap could also further alter
the expected light curves.

In the case of micro-TDEs in AGNs, a large fraction of the stellar mass
remains gravitationally bound to the BH, resulting in a bolometric luminosity
of up to $\sim 10^{44}$ erg s-1 (Kremer et al., 2019). By comparison, the
amount of stellar matter available in the case of TDEs depends on the
penetration parameter $\beta\equiv R_{\rm t}/R_{\rm p}$. Partial disruption
($\beta\lesssim 1$) is more common, resulting in lower luminosity (Stone et
al., 2020).

The long-term evolution of TDEs can be characterized by a smooth power-law
decay (van Velzen et al., 2021), with a fiducial power-law index of $-5/3$
(Rees, 1988). Micro-TDEs are expected to result in similar power-law decays in
the long term (Kremer et al., 2019). SPH simulations of micro-TDEs found the
power-law index to vary from $t^{-5/3}$ to $t^{-9/4}$, with steeper indices
corresponding to lower BH mass (Wang et al., 2021).

The high accretion rate in micro TDEs might also launch relativistic outflows
that produce significant $\gamma$-ray and X-ray emission. Such emission would
be weaker and longer than typical GRBs, possibly resembling that of ultralong
GRBs (Perets et al., 2016). Such $\gamma$ and X-ray emission is highly beamed
and is therefore only detectable from a fraction of micro TDEs. Micro TDE-
driven GRBs could be differentiated from other types of GRBs through the
identification of a thermal TDE-like counterpart with longer-wavelength
observations, and possibly directional coincidence with AGNs.

##  5 SMBHs too heavy for TDEs

For SMBHs whose Schwarzschild radius is greater than their tidal radius, stars
will be swallowed whole before they can be tidally disrupted. The maximum
(non-rotating) SMBH mass capable of producing TDEs, called the Hills mass
(Hills, 1975), is around $10^{8}$ M⊙ for solar-type stars (Stone et al.,
2020). This limit is only weakly dependent on the stellar mass for zero-age
main sequence stars, although it can be greater for off-main-sequence giant
stars, or for highly spinning SMBHs.

Micro-TDEs in AGNs are not limited by the SMBH mass. Therefore, given the
uncertainties in their light curves and spectra, the identification of TDE-
candidates in AGNs with SMBH mass beyond $10^{8}$ M⊙ may help distinguish
micro-TDEs and TDEs.

There has been two identifications of TDE-like emission from AGNs with SMBH
mass $M_{\bullet}>10^{8}$ M⊙:

ASASSN-15lh was an unusually bright transient first discovered by the All-Sky
Automated Survey for Supernovae (ASAS-SN; Shappee et al. 2014) in 2015 at a
redshift of $z=0.232$ (Nicholls et al., 2015). The spectrum of ASASSN-15lh
points to a TDE and rules out superluminous supernova origin (Leloudas et al.,
2016; Krühler et al., 2018). ASASSN-15lh was found to come from a galactic
nucleus with SMBH mass $M_{\bullet}=5^{+8}_{-3}\times 10^{8}$ M⊙ (Krühler et
al., 2018), which may be a low-luminosity AGN based on its BPT classification
(Krühler et al., 2018). This is beyond the Hills mass for solar-type stars,
although a highly spinning SMBH in this mass range could produce a TDE. The
late-time ($\gtrsim 100$ days) light curve and spectrum of ASASSN-15lh could
be well explained with a close to maximally spinning SMBH with mass
$M_{\bullet}\sim 10^{9}$ M⊙, although the early light curve and the lack of
radio and dim X-ray emission are not understood (Mummery & Balbus, 2020). The
peak luminosity of ASASSN-15lh was $L_{\rm peak}\sim 5\times 10^{45}$ erg s-1,
higher than the $\sim 10^{44}$ erg s-1 predicted from analytical micro-TDE
models (Leloudas et al., 2016), which nevertheless have large uncertainties.

ZTF19aailpwl was a bright ($L_{\rm peak}\sim 10^{45}$ erg s-1) transient first
observed by the Zwicky Transient Facility (Graham et al., 2019) in 2019 at a
redshift of $z=0.37362$ (Frederick et al., 2020). It originated from an AGN
with a reconstructed SMBH mass of $M_{\bullet}\sim 10^{8.2}$ M⊙ (although a
mass $<10^{8}$ M⊙ cannot be completely ruled out; Graham et al. 2019). Follow-
up observations found that the spectral properties of ZTF19aailpwl are
consistent with a TDE (Graham et al., 2019), but also with an AGN flare
attributed to enhanced accretion reported by Trakhtenbrot et al. (2019). Its
light curve is similar to that of TDEs, with a somewhat longer than usual rise
time. For micro-TDEs, such longer rise time may be possible due to interaction
with the disk wind and the low eccentricity of the stellar orbit that makes
the encounter less impulsive and the tidal effects more gradual, compared to
the usual SMBH-TDE case.

For both cases above, the reconstructed black hole masses beyond the Hills
mass point away from TDEs but are consistent with a micro-TDE origin.

Micro-TDEs can also occur outside of AGN disks, for example in stellar triples
hosting a BH (Fragione et al., 2019), or in dense stellar clusters (Kremer et
al., 2019; Samsing et al., 2019; Fragione et al., 2021). The rate of micro-
TDEs from these channels is uncertain but could be as high as that of TDEs.
These micro-TDEs could also occur in galaxies with SMBH mass above the Hills
mass. Observationally, however, these micro-TDE channels will occur far from
galactic centers and will not be confined to AGNs. As both ASASSN-15lh and
ZTF19aailpwl were localized to the galactic center and in an AGN means that,
for these two cases, non-AGN-assisted micro-TDE channels are unlikely.

##  6 Rate Density

To estimate the expected rate density of micro-TDEs, we computed the number of
stars, BHs and neutron stars that undergo orbital alignment with the AGN disk
within the AGN’s lifetime (assumed to be $10^{7}$ yr). We found $\sim 5$
orbital alignments for each object type (see also Yang et al. 2020c). We
simulated the merger of objects within the disk using a Monte Carlo simulation
of many galaxies, with random number of orbital alignments following a Poisson
distribution for each galaxy.

To compute the rate density we further assumed for simplicity that the
expected orbital alignment rate is identical in every AGN (there is only a
weak dependence on the AGN properties; Yang et al. 2019b). We adopted a number
density $n_{\rm seyfert}=0.018$ Mpc-3 for Seyfert galaxies (Hao et al., 2005)
which represent most AGNs. We show the resulting expected merger rate density
for all possible combinations of stars, BHs and neutron stars in Table 1.

| black hole | neutron star | star
---|---|---|---
black hole | 13 | 1 | 170
neutron star |  | 10-3 | 0.14
star |  |  | 20
Table 1: Expected rate density of binary encounters in AGNs. Micro-TDEs
correspond to BH–star encounters. Results are shown in units of Gpc-3 yr-1.
While BHs are the least common, they dominate the merger rate as their higher
mass results in more efficient mass segregation and orbital alignment, while
their mergers result in BHs that can undergo further mergers in the AGN disk.

We separately estimated the micro-TDE rate density from observations. For this
we used the rate density of TDE candidates in galaxies with
$M_{\bullet}\gtrsim 10^{8}$ M⊙, which are difficult to explain with disruption
by SMBHs. We used the TDE host galaxy SMBH mass function estimated by van
Velzen (2018) based on about two dozen observed TDE-candidates. van Velzen
(2018) found that the mass function is roughly constant for
$M_{\bullet}<10^{7.5}$ M⊙, while it sharply drops for $M_{\bullet}>10^{7.5}$
M⊙ (in this high mass range they only consider the detection of ASASSN-15lh).
Their reconstructed mass function corresponds to a TDE rate of
$5^{+8}_{-4}\times 10^{-2}$ Gpc-3yr-1 (1$\sigma$ uncertainty) for
$M_{\bullet}>10^{8}$ M⊙. Taking into account the mass function of SMBHs in
AGNs, we found that about $1\%$ of micro-TDEs occur in AGNs with
$M_{\bullet}>10^{8}$ M⊙, corresponding to a rate of $\sim 2$ Gpc-3yr-1.

##  7 Conclusion

We proposed that micro-TDEs occur in AGN disks, and found their rate to be
about $170$ Gpc-3yr-1. Such micro-TDEs may be easiest to distinguish from TDEs
around SMBHs by focusing on AGN-hosting galaxies in which the central SMBH’s
mass is too high ($M_{\bullet}\gtrsim 10^{8}$M⊙) to tidally disrupt solar-like
stars. Two such TDE candidates have been reported so far, ASASSN-15lh and
ZTF19aailpwl, both among the highest-luminosity TDE candidates.

The observed rate density of TDE candidates from galaxies with
$M_{\bullet}\gtrsim 10^{8}$M⊙ (van Velzen, 2018) is below the micro-TDE rate
density of $\sim 2$ Gpc-3yr-1 predicted in this work for AGNs with
$M_{\bullet}\gtrsim 10^{8}$M⊙, and might be a bright sub-population. In
addition, one candidate (ZTF19aailpwl) occurred in an AGN, while the other
(ASASSN-15lh) probably occurred in a weak AGN.

The unique environments in AGN disks are expected to give rise to a wealth of
other interesting phenomena, such as the tidal disruption of a star by a
neutron star (albeit at a smaller rate of $0.1$ Gpc-3yr-1), the formation of
Thorne-Zytkow objects (Thorne & Zytkow, 1975) when a neutron star in the disk
is unable to tidally disrupt a stellar companion due to a large stellar
radius, the tidal disruption of white dwarfs by stellar-mass BHs, with unique
signatures (Maguire et al., 2020), and the accretion induced collapse of
neutron stars (Perna et al., 2021b) and white dwarfs (Zhu et al., 2021a).

Further theoretical and observational work is needed to better understand the
spectral and temporal properties of AGN-assisted micro-TDEs, and to observe
them against the background of AGN variability. In particular we suggest an
observational focus on AGNs that harbor the heaviest SMBHs and exhibit unusual
flaring activity (e.g., Trakhtenbrot et al. 2019).

The authors would like to thank Sjoert van Velzen for useful suggestions. I.B.
acknowledges the support of the the Alfred P. Sloan Foundation. ZH was
supported by NASA grant NNX15AB19G and NSF grants AST-2006176 and AST-1715661.
HT acknowledges support by the Grants-in-Aid for Basic Research by the
Ministry of Education, Science and Culture of Japan (HT:17H01102, 17H06360).
RP acknowledges support by NSF award AST-2006839 and from NASA (Fermi) award
80NSSC20K1570.

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