#  Joint-Diagonalizability-Constrained Multichannel Nonnegative Matrix
Factorization Based on Multivariate Complex Sub-Gaussian Distribution
††thanks:  This work was partly supported by SECOM Science and
TechnologyFoundation and JSPS KAKENHI Grant Numbers JP19H01116, JP19H04131,
and JP19K20306, and JSPS-CAS Joint Research Program, Grant number
JPJSBP120197203.

Keigo Kamo1, Yuki Kubo1, Norihiro Takamune1, Daichi Kitamura2
Hiroshi Saruwatari1, Yu Takahashi3, Kazunobu Kondo3  1 Graduate School of
Information Science and Technology, The University of Tokyo, Tokyo 113-8656,
Japan  2 National Institute of Technology, Kagawa College, Kagawa 761-8058,
Japan  3 R&D Division, Yamaha Corporation, Shizuoka 430-8650, Japan

###### Abstract

In this paper, we address a statistical model extension of multichannel
nonnegative matrix factorization (MNMF) for blind source separation, and we
propose a new parameter update algorithm used in the sub-Gaussian model. MNMF
employs full-rank spatial covariance matrices and can simulate situations in
which the reverberation is strong and the sources are not point sources. In
conventional MNMF, spectrograms of observed signals are assumed to follow a
multivariate Gaussian distribution. In this paper, first, to extend the MNMF
model, we introduce the multivariate generalized Gaussian distribution as the
multivariate sub-Gaussian distribution. Since the cost function of MNMF based
on this multivariate sub-Gaussian model is difficult to minimize, we
additionally introduce the joint-diagonalizability constraint in spatial
covariance matrices to MNMF similarly to FastMNMF, and transform the cost
function to the form to which we can apply the auxiliary functions to derive
the valid parameter update rules. Finally, from blind source separation
experiments, we show that the proposed method outperforms the conventional
methods in source-separation accuracy.

###### Index Terms:

blind source separation, spatial covariance matrix, joint diagonalizability,
sub-Gaussian distribution

##  I Introduction

Blind source separation (BSS) [1] is a technique to separate sound sources
from observed mixtures without any prior information about the sources or
mixing system. For a determined or overdetermined situation, when the sources
are point sources and the reverberation is sufficiently short (referred to as
the rank-1 spatial model), frequency-domain independent component analysis [2,
3], independent vector analysis (IVA) [4, 5, 6], and independent low-rank
matrix analysis (ILRMA) [7, 8] have been proposed. However, the rank-1 spatial
model does not hold in the case of spatially spread sources or strong
reverberation.

Multichannel nonnegative matrix factorization (MNMF) [9, 10] is an extension
of nonnegative matrix factorization (NMF) [11] to multichannel cases, which
estimates the spatial covariance matrices of each source. MNMF employs full-
rank spatial covariance matrices [12], and this model can simulate situations
where, e.g., the reverberation is longer than the length of time-frequency
analysis. However, it has been reported that MNMF has a huge computational
cost and its performance strongly depends on the initial values of parameters
[7].

In the original MNMF, the observed signal is assumed to follow a time-variant
multivariate complex Gaussian distribution. Recently, the generative model
extension of MNMF to multivariate complex Student’s t distribution (t-MNMF
[13]) has been proposed. However, the original Gaussian MNMF and t-MNMF cannot
assume that the generative model follows a multivariate sub-Gaussian
distribution, whereas we reported that the separation accuracy is improved by
expanding the source signal model to the univariate sub-Gaussian distribution
in ILRMA (hereafter referred to as sub-Gaussian ILRMA) [14]. Thus, we can
expect that the introduction of sub-Gaussianity in MNMF leads to the
improvement of separation accuracy.

In this paper, we provide three contributions, namely, a generalization of the
generative model, its parameter optimization algorithm with a joint-
diagonalizability constraint, and experimental evaluation of the proposed
MNMF. First, we extend the generative model of MNMF to a time-variant
multivariate complex sub-Gaussian distribution; this is hereafter referred to
as sub-Gaussian MNMF. We employ a multivariate complex generalized Gaussian
distribution (GGD) as a sub-Gaussian distribution by restricting its shape
parameter. It is reported that some musical instrument signals obey sub-
Gaussian distributions [15]. Next, we derive parameter update rules of sub-
Gaussian MNMF. The cost function of sub-Gaussian MNMF is difficult to
minimize, and the auxiliary function for the majorization-minimiazation
algorithm [16] has not been discovered so far. To design the valid auxiliary
function, we introduce a joint-diagonalizability constraint in spatial
covariance matrices. This constraint has been introduced for the first time in
FastMNMF [17] to reduce the computational complexity of MNMF. On the other
hand, in this paper, we employ this joint-diagonalizability constraint not to
reduce the computational complexity but to transform the cost function to the
form to which we can apply the auxiliary function technique. To the best of
our knowledge, this is the world’s first update algorithm of parameters that
guarantees a monotonic nonincrease in the cost function of MNMF with a
multivariate complex sub-Gaussian model. Finally, we conduct BSS experiments
under reverberant conditions, showing that the proposed sub-Gaussian MNMF
outperforms conventional methods in source-separation accuracy.

##  II Conventional Methods

###  II-A MNMF [9, 10]

The short-time Fourier transform (STFT) of the observed multichannel signal is
defined as

$\displaystyle\bm{x}_{ij}=(x_{ij,1},\dots,x_{ij,M})^{\mathsf{T}}\in\mathbb{C}^{M},$
(1)

where $i=1,\dots,I,\,j=1,\dots,J,$ and $m=1,\dots,M$ are the indices of the
frequency bins, time frames, and channels, respectively, and
$\cdot^{\mathsf{T}}$ denotes the transpose. The MNMF model estimates a time-
variant parameter $\sigma_{ijn}$, which represents a character of the source,
and a time-invariant parameter $\bm{G}_{in}$, which represents spatial
characteristics of the source, where $n=1,\dots,N$ is the index of the
sources. $\sigma_{ijn}$ corresponds to a power spectrogram and $\bm{G}_{in}$
is called a spatial covariance matrix. In MNMF, as the generative model of the
observed signal $\bm{x}_{ij}$, the following multivariate complex Gaussian
distribution is assumed:

$\displaystyle\bm{x}_{ij}\sim\mathcal{N}\Bigl{(}\mathbf{0}_{M},\sum_{n}\sigma_{ijn}\bm{G}_{in}\Bigr{)},$
(2)

where $\mathbf{0}_{M}\in\mathbb{C}^{M}$ is an $M$-dimensional zero vector and
$\mathcal{N}(\bm{\mu},\bm{\Sigma})$ is the multivariate complex Gaussian
distribution whose mean is $\bm{\mu}$ and the covariance matrix is
$\bm{\Sigma}$. The source model $\sigma_{ijn}$ is a spectrogram of the $n$th
source at the $i$th frequency and $j$th time frame, having a low-rank spectral
structure represented by NMF, as

$\displaystyle\sigma_{ijn}=\sum_{k}t_{ik}v_{kj}z_{kn},$ (3)

where $k=1,\dots,K$ is the index of the NMF basis, and
$t_{ik}\in\mathbb{R}_{\geq 0}$ and $v_{kj}\in\mathbb{R}_{\geq 0}$ represent
the $i$th frequency component of the $k$th basis and the $j$th time-frame
activation component of the $k$th basis, respectively. In addition,
$z_{kn}\in\mathbb{R}_{\geq 0}$ is a latent variable that indicates whether the
$k$th basis belongs to the $n$th source. From (2), the negative log-likelihood
of the observed signal, which is a cost function to be minimized, is given by

$\displaystyle\mathcal{L}_{\mathrm{MNMF}}$
$\displaystyle\overset{c}{=}\sum_{i,j}\Bigl{(}\bm{x}_{ij}^{\mathsf{H}}\hat{\bm{X}}_{ij}^{-1}\bm{x}_{ij}+\log\det\hat{\bm{X}}_{ij}\Bigr{)},$
(4)

where
$\hat{\bm{X}}_{ij}=\sum_{n}\sigma_{ijn}\bm{G}_{in}=\sum_{k,n}t_{ik}v_{kj}z_{kn}\bm{G}_{in}$
and $\overset{c}{=}$ denotes equality up to a constant. We can estimate
$t_{ik},v_{kj},z_{kn},$ and $\bm{G}_{in}$ by minimizing (4). $\bm{G}_{in}$ can
be optimized by solving the Riccati equation and the remaining parameters are
updated by using the auxiliary function technique (details of these update
rules are described in [10]). After the update, we can estimate the separated
signal $\hat{\bm{s}}_{ijn}$ using the multichannel Wiener filter:

$\displaystyle\hat{\bm{s}}_{ijn}=\biggl{(}\sum_{k}t_{ik}v_{kj}z_{kn}\bm{G}_{in}\biggr{)}\hat{\bm{X}}_{ij}^{-1}\bm{x}_{ij}.$
(5)

MNMF assumes that $\bm{G}_{in}$ is a full-rank matrix [12], and this increases
versatility for various types of spatial condition. However, this full-rank
nature requires a large amount of computation.

###  II-B FastMNMF [17, 18]

To reduce the computational complexity of the update algorithm, FastMNMF
additionally assumes that the spatial covariance matrices
$\bm{G}_{i1},\dots,\bm{G}_{iN}$ are jointly diagonalizable by
$\bm{Q}_{i}=(\bm{q}_{i1},\dots,$ $\bm{q}_{iM})^{\mathsf{H}}$, which does not
depend on the source index $n$, as

$\displaystyle\bm{Q}_{i}\bm{G}_{in}\bm{Q}_{i}^{\mathsf{H}}=\bm{\mathcal{G}}_{in}\hskip
14.22636pt(n=1,\dots,N),$ (6)

where $\bm{\mathcal{G}}_{in}$ is a diagonal matrix. From (4) and (6), the
negative log-likelihood of the observed signal is given by

$\displaystyle\mathcal{L}_{\mathrm{F}}$
$\displaystyle\overset{c}{=}\sum_{i,j,m}\biggl{[}\frac{|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{2}}{\sum_{n,k}t_{ik}v_{kj}z_{kn}g_{inm}}+\log\sum_{n,k}t_{ik}v_{kj}z_{kn}g_{inm}\biggr{]}$
$\displaystyle\quad-2J\sum_{i}\log|\det\bm{Q}_{i}|,$ (7)

where $g_{inm}$ is the $m$th diagonal element of $\bm{\mathcal{G}}_{in}$. The
joint-diagonalization matrix $\bm{Q}_{i}$ in (II-B) can be optimized by
iterative projection (IP) [19], and the remaining parameters are updated by
using the auxiliary function technique [18]. Note that the algorithm of the
update of spatial covariance matrices in FastMNMF is different from that in
MNMF, which uses the Riccati equation. This algorithmic improvement provides a
different separation performance; FastMNMF is almost the same as or slightly
better than MNMF [18, 20]. After the update, we can estimate the separated
signal using the multichannel Wiener filter similarly to (5).

##  III Proposed Method

###  III-A Motivation and Strategy

In this paper, we propose a model extension of MNMF to the multivariate
complex sub-Gaussian distribution. Sub-Gaussian MNMF can appropriately model
the observed signal that follows the sub-Gaussian distribution, which cannot
be represented by the conventional methods. The multivariate GGD is used as
the sub-Gaussian distribution. The probability density function of the zero-
mean multivariate GGD is given by

$\displaystyle
p(\bm{x};\bm{0}_{M},\bm{\Sigma},\beta)=\frac{C(\beta)}{\det\bm{\Sigma}}\exp\Bigl{(}-(\bm{x}^{\mathsf{H}}\bm{\Sigma}^{-1}\bm{x})^{\beta/2}\Bigr{)},$
(8)

where $\beta>0$ is the shape parameter and $C(\beta)$ is the normalizing
constant of the multivariate GGD. In the case of $0<\beta<2$, the GGD becomes
super-Gaussian. In the case of $\beta>2$, the GGD becomes sub-Gaussian. For
$\beta=2$, the GGD corresponds to the Gaussian distribution. The negative log-
likelihood of MNMF based on the multivariate GGD model is represented as

$\displaystyle\mathcal{L}_{\mathrm{GGD}}\overset{c}{=}\sum_{i,j}\Bigl{(}(\bm{x}_{ij}^{\mathsf{H}}\hat{\bm{X}}_{ij}^{-1}\bm{x}_{ij})^{\beta/2}+\log\det\hat{\bm{X}}_{ij}\Bigr{)},$
(9)

where we substitute the model parameter $\hat{\bm{X}}_{ij}$ for $\Sigma$ in
(8).

In the case of $\beta>2$, this cost function (9) is difficult to minimize,
especially in the parameter $\bm{G}_{in}$ because it is impossible to design a
majorization function to which we can apply the Riccati equation solver (no
quadratic function can majorize the first term of the right-hand side of (9)).
Thus, we cannot derive the update rules of the sub-Gaussian MNMF that
guarantees a monotonic nonincrease in the cost function without any
constraint. Hence, we additionally introduce the joint-diagonalizability
constraint. If we apply the joint-diagonalizability constraint and an
appropriate auxiliary function to (9), we can find that the optimization
problem here is identical to a demixing-matrix-optimization problem in sub-
Gaussian ILRMA, which can be solved by the authors’ previous work, generalized
IP [14]. In the next subsections, details of the proposed parameter update
algorithm is described.

###  III-B Proposed Sub-Gaussian MNMF

By substituting (6) into (9), we obtain the cost function of MNMF with the
multivariate GGD model as

$\displaystyle\mathcal{L}_{\mathrm{GGD}}$
$\displaystyle\overset{c}{=}-2J\sum_{i}\log|\det\bm{Q}_{i}|+\sum_{i,j,m}\log\sum_{n,k}t_{ik}v_{kj}z_{kn}g_{inm}$
$\displaystyle\phantom{=}+\sum_{i,j}\biggl{(}\sum_{m}\frac{|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{2}}{\sum_{n,k}t_{ik}v_{kj}z_{kn}g_{inm}}\biggr{)}^{\frac{\beta}{2}}.$
(10)

First, we derive update rules of parameters $t_{ik},v_{kj},z_{kn},$ and
$g_{inm}$. For $\beta>2$, $f_{1}(y)=y^{\beta/2}$ is a convex function. Hence,
by applying Jensen’s inequality to (III-B), we have

$\displaystyle\mathcal{L}_{\mathrm{GGD}}$
$\displaystyle\overset{c}{=}\sum_{i,j}\biggl{(}\sum_{m}\frac{|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{2}}{\sum_{n,k}t_{ik}v_{kj}z_{kn}g_{inm}}\biggr{)}^{\frac{\beta}{2}}$
$\displaystyle\phantom{=}+\sum_{i,j,m}\log\sum_{n,k}t_{ik}v_{kj}z_{kn}g_{inm}$
$\displaystyle\overset{c}{\leq}\sum_{i,j}\sum_{m}\tilde{\xi}_{ijm}\biggl{(}\frac{|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{2}}{\tilde{\xi}_{ijm}\sum_{n,k}t_{ik}v_{kj}z_{kn}g_{inm}}\biggr{)}^{\frac{\beta}{2}}$
$\displaystyle\phantom{=}+\sum_{i,j,m}\log\sum_{n,k}t_{ik}v_{kj}z_{kn}g_{inm},$
(11)

where $\tilde{\xi}_{ijm}\geq 0$ is an auxiliary variable that satisfies
$\sum_{m}\tilde{\xi}_{ijm}=1$, and $\overset{c}{\leq}$ denotes that the left-
hand side is less than or equal to the right-hand side up to a constant.
Moreover, we can design an auxiliary function of $f_{2}(y)=(1/y)^{\beta/2}$ by
applying Jensen’s inequality because it is also a convex function. We can also
design one of $f_{3}(y)=\log y$ by applying the tangent inequality because it
is a concave function. Then, we can obtain the final auxiliary function
$\mathcal{L}^{+}_{\mathrm{Sub}}$ as

$\displaystyle\mathcal{L}_{\mathrm{GGD}}$
$\displaystyle\overset{c}{\leq}\sum_{i,j,m}(\tilde{\xi}_{ijm})^{1-\frac{\beta}{2}}\sum_{n,k}\tilde{\eta}_{ijkmn}\biggl{(}\frac{\tilde{\eta}_{ijkmn}|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{2}}{t_{ik}v_{kj}z_{kn}g_{inm}}\biggr{)}^{\frac{\beta}{2}}$
$\displaystyle\phantom{=}+\sum_{i,j,m}\frac{1}{\tilde{\zeta}_{ijm}}\sum_{n,k}t_{ik}v_{kj}z_{kn}g_{inm}$
(12) $\displaystyle:=\mathcal{L}^{+}_{\mathrm{Sub}},$ (13)

where $\tilde{\eta}_{ijkmn}\geq 0,\tilde{\zeta}_{ijm}\geq 0$ are auxiliary
variables, and $\tilde{\zeta}_{ijm}$ satisfies
$\sum_{m}\tilde{\zeta}_{ijm}=1$. The equalities of (III-B) and (III-B) hold if
and only if the auxiliary variables are set as follows:

$\displaystyle\tilde{\xi}_{ijm}$
$\displaystyle=\left.\frac{|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{2}}{\sum_{k,n}t_{ik}v_{kj}z_{kn}g_{inm}}\middle/\sum_{m^{\prime}}\frac{|\bm{q}_{im^{\prime}}^{\mathsf{H}}\bm{x}_{ij}|^{2}}{\sum_{k,n}t_{ik}v_{kj}z_{kn}g_{inm^{\prime}}}\right.,$
(14) $\displaystyle\tilde{\eta}_{ijkmn}$
$\displaystyle=\frac{t_{ik}v_{kj}z_{kn}g_{inm}}{\sum_{k,n}t_{ik}v_{kj}z_{kn}g_{inm}},$
(15) $\displaystyle\tilde{\zeta}_{ijm}$
$\displaystyle=\sum_{k,n}t_{ik}v_{kj}z_{kn}g_{inm}.$ (16)

The update rules for (III-B) w.r.t. $t_{ik},v_{kj},z_{kn},$ and $g_{inm}$ are
derived by setting the gradient to zero. From
$\partial\mathcal{L}_{\mathrm{Sub}}^{+}/\partial t_{ik}=0$,
$\partial\mathcal{L}_{\mathrm{Sub}}^{+}/\partial v_{kj}=0$,
$\partial\mathcal{L}_{\mathrm{Sub}}^{+}/\partial z_{kn}=0$, and
$\partial\mathcal{L}_{\mathrm{Sub}}^{+}/\partial g_{inm}=0$, we obtain

$\displaystyle\chi_{ijm}$ $\displaystyle=\sum_{k,n}t_{ik}v_{kj}z_{kn}g_{inm},$
(17) $\displaystyle\phi_{ijm}$
$\displaystyle=|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{2}\biggl{(}\sum_{m^{\prime}}\frac{|\bm{q}_{im^{\prime}}^{\mathsf{H}}\bm{x}_{ij}|^{2}}{\chi_{ijm^{\prime}}}\biggr{)}^{\frac{\beta-2}{2}},$
(18) $\displaystyle t_{ik}$ $\displaystyle\leftarrow
t_{ik}\Biggl{(}\frac{\beta\sum_{j,n,m}\frac{\phi_{ijm}v_{kj}z_{kn}g_{inm}}{\chi_{ijm}^{2}}}{2\sum_{j,n,m}\frac{v_{kj}z_{kn}g_{inm}}{\chi_{ijm}}}\Biggr{)}^{\frac{2}{\beta+2}},$
(19) $\displaystyle v_{kj}$ $\displaystyle\leftarrow
v_{kj}\Biggl{(}\frac{\beta\sum_{i,n,m}\frac{\phi_{ijm}t_{ik}z_{kn}g_{inm}}{\chi_{ijm}^{2}}}{2\sum_{i,n,m}\frac{t_{ik}z_{kn}g_{inm}}{\chi_{ijm}}}\Biggr{)}^{\frac{2}{\beta+2}},$
(20) $\displaystyle z_{kn}$ $\displaystyle\leftarrow
z_{kn}\Biggl{(}\frac{\beta\sum_{i,j,m}\frac{\phi_{ijm}t_{ik}v_{kj}g_{inm}}{\chi_{ijm}^{2}}}{2\sum_{i,j,m}\frac{t_{ik}v_{kj}g_{inm}}{\chi_{ijm}}}\Biggr{)}^{\frac{2}{\beta+2}},$
(21) $\displaystyle g_{inm}$ $\displaystyle\leftarrow
g_{inm}\Biggl{(}\frac{\beta\sum_{j,k}\frac{\phi_{ijm}t_{ik}v_{kj}z_{kn}}{\chi_{ijm}^{2}}}{2\sum_{j,k}\frac{t_{ik}v_{kj}z_{kn}}{\chi_{ijm}}}\Biggr{)}^{\frac{2}{\beta+2}}.$
(22)

Note that these update rules have already been substituted under the equality
conditions (14)–(16) and rearranged.

Next, we derive the update rule of the joint-diagonalization matrix
$\bm{Q}_{i}$. We can design the auxiliary function of the cost function
(III-B) w.r.t. $\bm{Q}_{i}$ similarly to the derivation of (III-B) as

$\displaystyle\mathcal{L}_{\mathrm{GGD}}$
$\displaystyle\overset{c}{\leq}\sum_{i,j,m}\tilde{\xi}_{ijm}\biggl{(}\frac{|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{2}}{\tilde{\xi}_{ijm}\sum_{n,k}t_{ik}v_{kj}z_{kn}g_{inm}}\biggr{)}^{\frac{\beta}{2}}$
$\displaystyle\phantom{=}-2J\sum_{i}\log|\det\bm{Q}_{i}|$
$\displaystyle=\sum_{i,j,m}\frac{|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{\beta}}{r_{ijm}^{\beta}}-2J\sum_{i}\log|\det\bm{Q}_{i}|,$
(23)

where

$\displaystyle
r_{ijm}=\tilde{\xi}_{ijm}^{\frac{1}{2}-\frac{1}{\beta}}\biggl{(}\sum_{k,n}t_{ik}v_{kj}z_{kn}g_{inm}\biggr{)}^{\frac{1}{2}}.$
(24)

The auxiliary function (23) is the same form as that for the optimization of a
demixing matrix in sub-Gaussian ILRMA [14]. Hence, we derive the update rule
in the same manner as sub-Gaussian ILRMA. Similarly to [14], in the case of
$2<\beta\leq 4$, we can bound the term
$|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{\beta}$ using the inequality of
weighted arithmetic and geometric means as follows:

$\displaystyle|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{\beta}\leq\frac{\beta}{4}\frac{|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{4}}{\tilde{\alpha}_{ijm}^{4-\beta}}+\Bigl{(}1-\frac{\beta}{4}\Bigr{)}\tilde{\alpha}_{ijm}^{\beta},$
(25)

where $\tilde{\alpha}_{ijm}$ is an auxiliary variable and the equality of (25)
holds if and only if
$\tilde{\alpha}_{ijm}=|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|$. We can apply
(25) to (23) and obtain

$\displaystyle\mathcal{L}_{\mathrm{GGD}}\overset{c}{\leq}\frac{\beta}{4}\sum_{i,j,m}\frac{|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{4}}{\tilde{\alpha}_{ijm}^{4-\beta}r_{ijm}^{\beta}}-2J\sum_{i}\log|\det\bm{Q}_{i}|.$
(26)

Moreover, we can design a further auxiliary function of (26) as

$\displaystyle\mathcal{L}_{\mathrm{GGD}}\leq
J\sum_{i,n}(\bm{q}_{im}^{\mathsf{H}}\bm{B}_{im}\bm{q}_{im})^{2}-2J\sum_{i}\log|\det\bm{Q}_{i}|,$
(27)

where $\bm{B}_{im}$ is an auxiliary variable that satisfies

$\displaystyle l_{ijm}$
$\displaystyle=\sqrt[4]{|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{4-\beta}r_{ijm}^{\beta}},$
(28) $\displaystyle\bm{H}_{im}$
$\displaystyle=\biggl{[}\frac{1}{l_{i1m}}\bm{x}_{i1}\cdots\frac{1}{l_{iJm}}\bm{x}_{iJ}\biggr{]},$
(29) $\displaystyle\bm{a}_{im}$ $\displaystyle=\biggl{[}a_{i1m}\cdots
a_{iJm}\biggr{]}^{\mathsf{T}}=\bm{H}_{im}^{\mathsf{H}}\tilde{\bm{q}}_{im},$
(30) $\displaystyle\bm{A}_{im}$
$\displaystyle=\begin{bmatrix}||\bm{a}_{im}||^{2}&-a_{i1m}\overline{a_{i2m}}&\cdots&-a_{i1m}\overline{a_{iJm}}\\\
-a_{i2m}\overline{a_{i1m}}&||\bm{a}_{im}||^{2}&\cdots&-a_{i2m}\overline{a_{iJm}}\\\
\vdots&\vdots&\ddots&\vdots\\\
-a_{iJm}\overline{a_{i1m}}&-a_{iJm}\overline{a_{i2m}}&\cdots&||\bm{a}_{im}||^{2}\end{bmatrix},$
(31) $\displaystyle\bm{B}_{im}$
$\displaystyle=\frac{\sqrt{\beta}}{2\sqrt{J\sum_{j}|a_{ijm}|^{4}}}\bm{H}_{im}\bm{A}_{im}\bm{H}_{im}^{\mathsf{H}}.$
(32)

Here, $\overline{\cdot}$ denotes the complex conjugate and
$\tilde{\bm{q}}_{im}$ is an auxiliary variable and the equality of (27) holds
if and only if $\tilde{\bm{q}}_{im}=\bm{q}_{im}$. The joint-diagonalization
matrix $\bm{Q}_{i}$ of the cost function (27) can be updated by generalized IP
[14]. The update rule of $\bm{Q}_{i}$ can be derived as

$\displaystyle r_{ijm}$
$\displaystyle\leftarrow|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{1-\frac{2}{\beta}}\chi_{ijm}^{\frac{1}{\beta}}(\sum_{m^{\prime}}\frac{|\bm{q}_{im^{\prime}}^{\mathsf{H}}\bm{x}_{ij}|^{2}}{\chi_{ijm^{\prime}}})^{\frac{1}{\beta}-\frac{1}{2}}$
(33) $\displaystyle\bm{U}_{im}$
$\displaystyle=\sum_{j}\frac{\bm{x}_{ij}\bm{x}_{ij}^{\mathsf{H}}}{\sqrt{|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{4-\beta}r_{ijm}^{\beta}}},$
(34) $\displaystyle\bm{B}_{im}^{\prime}$
$\displaystyle=\bm{q}_{im}^{\mathsf{H}}\bm{U}_{im}\bm{q}_{im}\bm{U}_{im}+\sum_{j}\frac{|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{\beta-2}}{r_{ijm}^{\beta}}\bm{x}_{ij}\bm{x}_{ij}^{\mathsf{H}}$
$\displaystyle\phantom{=}-\Bigl{(}\bm{U}_{im}\bm{q}_{im}\Bigr{)}\Bigl{(}\bm{U}_{im}\bm{q}_{im}\Bigr{)}^{\mathsf{H}},$
(35) $\displaystyle\bm{q}_{im}$
$\displaystyle\leftarrow(\bm{Q}_{i}\bm{B}_{im}^{\prime})^{-1}\bm{e}_{m},$ (36)
$\displaystyle\bm{q}_{im}$
$\displaystyle\leftarrow\bm{q}_{im}\biggl{(}\frac{2J}{\beta\sum_{j}(|\bm{q}_{im}^{\mathsf{H}}\bm{x}_{ij}|^{\beta}/r_{ijm}^{\beta})}\biggr{)}^{\frac{1}{\beta}},$
(37)

where $\chi_{ijm}$ is defined in (17), and $\bm{B}_{im}^{\prime}$ is equal to
$\bm{B}_{im}$ up to scale. From (19)–(22) and (33)–(37), all of the parameters
$t_{ik},v_{kj},z_{kn},g_{inm}$, and $\bm{Q}_{i}$ can be iteratively updated in
the proposed sub-Gaussian MNMF.

##  IV Experiments

###  IV-A Experimental Conditions

We confirmed the efficacy of the proposed method by conducting music source
separation experiments. We compared six methods: IVA [19], ILRMA [7], sub-
Gaussian ILRMA [14], MNMF [10], FastMNMF [17], and the proposed sub-Gaussian
MNMF. We used monaural dry music sources of four melody parts [21, 22]. Eight
combinations of instruments with different melody parts were selected as shown
in Table I.

TABLE I: Dry sources used in experiment
---
| Part name | Source (1/2)
Music 1 | Midrange/Melody 2 | Flute/Piano
Music 2 | Melody 1/Melody 2 | Flute/Oboe
Music 3 | Melody 2/Midrange | Harpsichord/Violin
Music 4 | Melody 2/Bass | Cello/Violin
Music 5 | Melody 1/Bass | Cello/Oboe
Music 6 | Melody 2/Melody 1 | Trumpet/Violin
Music 7 | Bass/Melody 2 | Flute/Bassoon
Music 8 | Bass/Melody 1 | Trumpet/Bassoon

To simulate reverberant mixing, two-channel mixed signals were produced by
convoluting the impulse response E2A ($T_{60}=$ 300 ms) in the RWCP database
[23]. Fig. 1 shows the recording conditions of E2A used in our experiments. In
these mixtures, the input signal-to-noise ratio was 0 dB.

Figure 1: Spatial arrangements of sources and microphones.

The sampling frequency was 16 kHz and an STFT was performed using a 64 ms
Hamming window with a 16 ms shift ($T_{60}$ was longer than the window length,
i.e., the spatial covariance matrices were full rank). The total number of
bases in the low-rank source model was $K=20,30,40,50,$ and $60$. The
initializations of the source model parameters ($t_{ik},v_{kj},z_{kn}$) and
the spatial covariance matrix $\bm{G}_{in}$ were random values and the
identity matrix, respectively. The initialization of $\bm{Q}_{i}$ was the
identity matrix. The shape parameter $\beta$ in sub-Gaussian ILRMA and the
proposed sub-Gaussian MNMF was set to 4. The number of iterations in all
methods was 6000. We used the source-to-distortion ratio (SDR) improvement
[24] to evaluate the total separation performance.

###  IV-B Experimental Results

Figure 2: Resultant SDR improvement for each method. (a) Results of recording
condition (a) in Fig. 1. (b) Results of recording condition (b) in Fig. 1.

Fig. 2 shows the average SDR improvements over the source pairs and 10-trial
initialization. Compared with IVA, ILRMA, and sub-Gaussian ILRMA, both
conventional MNMF and FastMNMF as well as the proposed sub-Gaussian MNMF
provide better SDR improvements. This is because the rank-1 spatial model
(IVA, ILRMA, and sub-Gaussian ILRMA) does not hold in the case of strong
reverberation. The conventional full-rank spatial models (MNMF and FastMNMF)
achieve almost the same SDR improvements. On the other hand, the proposed sub-
Gaussian MNMF markedly outperforms the conventional full-rank spatial model
methods, regardless of the spatial arrangement. This suggests that the sub-
Gaussian model is more appropriate than the conventional Gaussian model for
simulating music signals, showing the effectiveness of the proposed sub-
Gaussian MNMF.

##  V conclusion

In this paper, we proposed the model extension of MNMF to the multivariate
complex sub-Gaussian distribution. We introduced the joint-diagonalizability
constraint in spatial covariance matrices to derive update rules that
guarantees a monotonic nonincrease in the cost function. From the source-
separation experiments, we showed that the proposed method outperformed the
conventional methods in terms of SDR improvement.

## References

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# The relationship between Biological and Artificial Intelligence

George Čevora PhD
Chief Data Scientist, illumr Ltd.
<EMAIL_ADDRESS>
###### Abstract

Intelligence can be defined as a predominantly human ability to accomplish
tasks that are generally hard for computers and animals. Artificial
Intelligence [AI] is a field attempting to accomplish such tasks with
computers. AI is becoming increasingly widespread, as are claims of its
relationship with Biological Intelligence. Often these claims are made to
imply higher chances of a given technology succeeding, working on the
assumption that AI systems which mimic the mechanisms of Biological
Intelligence should be more successful.

In this article I will discuss the similarities and differences between AI and
the extent of our knowledge about the mechanisms of intelligence in biology,
especially within humans. I will also explore the validity of the assumption
that biomimicry in AI systems aids their advancement, and I will argue that
existing similarity to biological systems in the way Artificial Neural
Networks [ANNs] tackle tasks is due to design decisions, rather than inherent
similarity of underlying mechanisms. This article is aimed at people who
understand the basics of AI (especially ANNs), and would like to be better
able to evaluate the often wild claims about the value of biomimicry in AI.

###### Acknowledgements

I thank Kate Wilkinson for extensive editing and proofreading of this article.
Dr Kristjan Kalm’s critical review of this article was essential for its
scientific integrity; however, not all views expressed in this article reflect
Kristjan’s own views. Finally, I would like to thank illumr Ltd, especially
its founder and CEO Jason Lee, for sponsoring this work.

##  1 From Symbolic AI to Machine Learning

Symbolic AI was the prevailing approach to AI until the early 90’s. It is
reliant on human programmers coding complex rules to enable machines to
complete complex tasks. Continuing failure of this approach to solve many
tasks crucial to intelligence provides a good contrast with Machine Learning –
an alternative approach to AI which is essential to the current advent of
artificially intelligent machines.

In 1994 the reigning chess champion Garry Kasparov was beaten by Deep Blue.
This was a great success for IBM at the time, and also for Artificial
Intelligence. However, Deep Blue, while capable of playing world-class chess,
could not move the pieces – a human helper was required for this task. We will
start the exploration of the relationship between Artificial and Biological
Intelligence by investigating the distinction between the tasks of deciding
about chess moves and physically moving the pieces.

While chess is generally considered a difficult game, its rules are actually
very simple and can easily fit on a single A4. Deep Blue was programmed to
calculate a lot of possible moves ahead according to those rules. Out of those
moves Deep Blue just picked the most beneficial one. The reason why it took
until 1994 for a computer to beat a human chess champion was a limitation on
computing power. While the rules of chess are simple, there are very many
possible moves - exploring even a small fraction of these requires a
gargantuan amount of computation111The success of Deep Blue was actually for a
significant part also due to making decisions which lines of game to avoid
exploring. This fact doesn’t have any bearing on the present argument though..
Computers were not able to make this many calculations within a realistic
time-frame before Deep Blue, but these days the computational power of Deep
Blue is roughly matched by an average smartphone. Because the rules are so
simple anyone with elementary coding skills can write a chess program and
given sufficient computational power can beat world chess champion simply by
applying the rules. This approach is called symbolic AI because the computer
arrives to a solution by following predefined rules (commonly called symbolic
computation).

Unfortunately, not all problems lend themselves to being solved in this way.
Moving a chess piece is a good example – I encourage you to stop reading for a
moment and think about what rules you would have to apply to select and move a
chess piece. You have to start from the beginning and recognize where the
piece is currently located - a standard task for image recognition systems
that are gaining so much traction these days. What does a chess piece look
like? Can you define rules that will allow the computer to recognize chess
pieces from any angle under any light conditions? How about pieces with novel
shapes that humans can instantaneously recognize? Consider the variety of
chess pieces in figure 1. The difficulty of making these definitions is the
reason why symbolic AI ultimately failed to translate to “easy” tasks such as
image recognition despite the huge success in “hard” tasks such as chess. At
the same time the hard-coded rules of symbolic AI do not bear any resemblance
to the fuzzy nature of biological intelligence. Machine Learning-based
approaches bear more resemblance to human intelligence than those with
symbolic AI, but is this really the driving force behind its huge success?

|
---|---
|
Figure 1: Humans can intuitively recognize a number of different styles of
previously unencountered chess pieces. This task is impossible for symbolic AI
because coding explicit rules for this purpose is impossible. This example is
in striking contrast with the game of chess rules of which can be summarised
on a single A4.

Machine Learning [ML], as the name suggests, is a branch of AI in which the
machines learn to solve a problem themselves, as opposed to being given a set
of explicit rules on how to solve the task. This is usually achieved by
learning new rules through repeated attempts at the task, and feedback on the
extent of success each time. This is the first fundamental similarity between
the current progress in AI and biological Intelligence. We were not born with
ability to recognize the objects surrounding us, neither (for the most part)
were we told rules allowing us to identify those objects. Instead we have seen
a very large number of objects that were sometimes labelled (think parents
pointing out objects to a child) and learned from the experience.

Superficially, Machine Learning works in a very similar way. Computers are
shown millions of pictures featuring various objects until they gradually
learn what dogs, hats, cats etc. look like. Despite that this similarity may
seem trivial and shallow it is by far the most important principle driving the
current success of AI. While the similarity between artificial and biological
intelligence goes much deeper, important differences remain. The machinery
that has allowed computers to learn these powerful rules – Artificial Neural
Networks [ANNs] - is directly inspired by neural networks in biological
brains. In the rest of this article we will investigate how relevant this
resemblance is to the abilities of this technology, and how deeply these
similarities really go.

##  2 Neurons

Figure 2: Basic structure of a Biological Neuron. A number of simplifications
were made. For a proper description, please refer to a contemporary
Neurobiology textbook.

The tasks accomplished by ANNs resemble those accomplished by biological
intelligence to some degree, and so do the building blocks of these two types
of intelligent systems. (A basic description of a Biological Neuron is
provided in figure 2 for reference purposes.) The similarities and differences
between artificial and biological neurons have been used to argue for
greater/lesser validity of methods in artificially replicating biological
intelligence. However, this view is greatly simplified. In this section I will
critically assess some of these claims one by one; however, the fundamental
conclusion I will argue for is that the similarities between artificial and
biological neurons are not a valid line of reasoning about the effectiveness
of AI.

In a nutshell, both types of neuron integrate the signal arriving from
multiple other neurons, transform it with a non-linear function and output it
to further neurons.

Many people rightly point out that biological neurons operate using discrete
signalling, while the artificial ones pass continuous numerical values.
Indeed, the action potential is binary - it is not known to have any
graduation between on and off. However, individual action potential spikes do
not usually significantly affect a biological neuron. Instead the spiking rate
(frequency) seems to be important. Frequency, being a continuous value itself,
doesn’t fundamentally differ from the information passed by artificial
neurons. Therefore the validity of this difference is called into question.

The similarity of the type of non-linearity applied by both types of neurons
is open to question too. It is true that the binary activation function (step
function) of biological neurons is poorly replicated by the sigmoidal non-
linearities traditionally used in ANNs. However, if we consider the value
output by artificial neurons to be akin to the firing frequency of biological
ones, then we must instead compare the function which results from a period of
firing. The superposition of many iterations of a fuzzy step function is well
represented by a sigmoidal non-linearity, suggesting that there may actually
be some degree of similarity in the activation function.

A large boost to the performance of ANNs was achieved not long ago by the
introduction of a novel activation function. The rectified linear function,
also called ReLU [7], helps neurons learn in situations that would formerly
not be possible, due to its elegant mathematical properties. The authors of
this discovery also praised ReLU for its biological plausibility. The logic
behind that argument was never clear to me. While I do agree that the
sigmoidal activation function doesn’t follow the activation function of
biological neurons when taken at the face value, ReLU doesn’t even stay in a
biologically plausible range. Instead it never saturates - meaning that more
input always means more output without any limit. Biological neurons have a
natural upper bound on how much they can fire - they get tired or even die
when firing too much.

For a long time it has been thought that a neuron performs a single signal
processing operation; however, more recently neuroscientists have revealed
that the integration and simple logical processing of signal actually happens
many times within a single neuron222The processing was found to happen already
at the dendrite - before signals from many dendrites merge - as opposed to
being limited only to the cell body (cf. Figure 2) [17]. This finding makes
the computation performed by a neuron several orders of magnitude more complex
than it was thought to be just few years ago. However, the computation is not
different, it is just performed multiple times within one neuron. Using
multiple artificial neurons where formerly a single one was used should
therefore address this issue.

Neurons have been also found to be sensitive to temporal patterns in their
stimulation [5] in a somewhat similar way to Morse code, which codes
information as a binary signal in time. While artificial neurons do not have
any such capability, any n-bit temporal pattern can be equally represented by
a static pattern across n neurons. Therefore, replicating this finding only
increases the quantity of the artificial neurons needed rather than their
individual complexity.

While a significant number of arguments highlighting the differences between
artificial and biological neurons do not stand up to scrutiny there are
certainly many aspects of biological neurons that are not replicated to any
degree by artificial neurons. Biological neurons get tired or even die when
firing for too long, their function is modulated by hormones, they recycle
neurotransmitters, and they are affected by concentration of salt and
potassium etc. The complexity is staggering and we are still far from
understanding how biological neurons work.

In fact, only a fraction of computation is performed by neurons. For decades,
glial cells that make-up roughly three quarters of the brain were thought to
be simply providing supporting function to neurons in the brain. However, they
have been discovered to receive, process and distribute signals in a way only
neurons were previously known to do [10]. As of yet the implications of the
computation performed by glial cells is very far from understood.

Furthermore, biologists like Dennis Bray have long argued that computation is
not unique to brains, but rather omnipresent in living organisms, and that
protein synthesis and other cellular processes can’t be denied the status of
computation. However, these processes are limited to within a cell and cannot
really affect the behaviour of an organism directly (despite being crucial to
its functionality), therefore this should have little effect on our
understanding of intelligence. This view is, however, being challenged by
current research.

Last year a series of discoveries caused a stir in scientific community when
fruit fly cells were found to synthesise virus-like proteins, and use them to
communicate with other cells [15]. While it’s unclear whether similar
phenomena exists in humans and the speed of such communication is likely
extremely low, it is an example of computation within a cell other than a
neuron being communicated across an organism. This is something we used to
think was special about neurons, or later about the brain. In the light of the
immense complexity of computation and communication in living organisms it
seems futile to model the neuron to any degree of veracity hoping that it will
provide us with technical advances.

Artificial neurons will for a foreseeable future still be a very inaccurate
models of biological neurons, which in turn are only a part of the machinery
used to perform computation and communication necessary for biological
intelligence. Does this mean that AI achieved by ANNs cannot in principle
achieve similar capabilities to biological intelligence?

Structurally, the artificial neuron is an extremely limited model for
biological neuron. However, more detailed models of neuron (e.g. Numenta) or
neural networks that communicate by discrete spiking (e.g. SpiNNaker), failed
profoundly to deliver any advantage to AI. Structural similarity between
neurons may therefore not be nearly as important property for AI as the
function of a neuron. Indeed as a basic unit performing elementary logical
processing, the two types of neuron could be seen as very similar.

I have always understood the neural networks as universal function
approximators whether they are biological or artificial (what else than an
approximator could they be!). The precise structure of a function approximator
cannot influence its computational capability as long as it is sufficiently
general to approximate the desired function and possible to train in finite
time. Given a sufficient number of neurons, ANNs can approximate any function
[11]. Therefore, there is no reason why they could not approximate a function
performed by biological neurons. This is equally true about any other
universal function approximator such as polynomials which do not aspire to any
similarity with biological neurons. The reason why ANNs are gaining much more
traction in AI than polynomials is that they are relatively easy to train,
this however may not hold for long as novel research suggests that polynomials
can beat ANNs on a number of ML benchmark problems [19]. The change that
brought about the advancement of polynomials was not the change of their
structure or computational capability, but novel methods of training them to
perform the desired function. In summary, it is in theory possible for ANNs to
approximate biological functionality without similarity in the basic building
blocks. The remaining question is whether we can train ANNs to perform the
function of our interest.

##  3 Networks and Learning

So far we have discussed only rudimentary properties of individual neurons.
However, the way neurons are put together into networks and trained has
historically had a larger impact on their performance than the structure of a
single unit.

The first practical ANN was introduced by Frank Rosenblatt in the 1950s,
called a Perceptron. It did not include any hidden layers and as such was
profoundly limited. However, its parallel nature allowed it to easily learn
simple logic - performing logic on parallel inputs is much easier than on
serial, as no memory is required to store intermediate values. Deep layers
were not around yet, mostly because no one knew how to train them. Minsky and
Pappert demonstrated in 1969 that this basic kind of ANN architecture can
never solve linearly inseparable problems (e.g. perform logical operations
like exclusive OR) and concluded that they will never be able to solve any
actually interesting problem. In their book they wrongly conjured that this
problem is common to all ANNs, which caused a huge decrease of
interest/funding in ANNs, essentially halting the research in the area for
almost two decades.

It took until the 1980s to salvage what was left of ANNs. An international
group of researchers (David Rumelhart, Jeff Hinton and others) came up with a
method to train deep ANNs called backpropagation. This immediately disproved
the conjecture of Minsky and Pappert and kicked-off a large research effort
into ANNs. It is worthwhile to note that backpropagation is perhaps the aspect
of ANNs most widely criticised for being biologically implausible.

Backpropagation relies on propagating errors backwards through a deep ANN to
correct/train the deep layers. The algorithm of backpropagation roots in
automatic differentiation, a method developed by Newton at a time when there
was essentially no understanding of the biology underlying intelligence.
Claiming biological inspiration as its inception would therefore be silly.

The main criticism of the biological plausibility of backpropagation is
focused on the requirement to feed signals backwards, which biological neurons
are known to be unable to do333This is not entirely true as some marginal
cases of backwards signalling have been reported.. However, neuroscientists
have known for a long time that neurons are not a uniform mass of homogeneous
material, but instead they repeatedly appear in structured assemblies that
have feedback connections [2]. Therefore it is entirely plausible that at
least in some brain areas neurons exist in pairs where one feeds forwards
while the other one is feeding back making backpropagation entirely possible
within a neural assembly. This being said, there has been no convincing
demonstration of backpropagation of error in brain so far - something we
should expact to be relatively easy to observe if it is a prominent feature of
biological neural networks.

Backpropagation was a necessary advancement for deep neural networks, as the
depth is one of the most powerful concepts brought to ANNs. Depth allows for
hierarchical representation of a problem - solving smaller problems (e.g.
detection of edges) first before moving onto bigger ones (e.g. recognizing an
object from a collection of edges). Hierarchical nature is also a prominent
feature of information processing in brains [16], providing an elegant
alignment between biological and artificial intelligence.

Another large improvement to the capabilities of ANNs was the introduction of
convolution as one of the main components [13]. Convolution essentially means
moving a filter across the data to identify features. This brought a large
advantage to the training of ANNs as it hugely decreases the number of free
parameters to be tweaked. Interestingly, it is well evidenced that similar
principles are being applied in the brain [6], and is perhaps are one of the
crucial phenomena allowing brain to process visual stimuli very efficiently.

So far we have seen that the criticism of backpropagation for it’s biological
implausibility is shaky (while no evidence for backpropagation within
biological neural networks has been found so far), and that convolution seems
biologically well founded. However, the last aspect of contemporary ANNs does
not follow this line of reasoning. Initialization of weights by an identity
matrix [12] is a technique that facilitates the training of deeper networks.
It works because initially the deep layers just pass on their inputs unaltered
– a better starting point than learning from a random complex transformation.
As you might expect this doesn’t lend itself to any meaningful comparison with
the brain, illustrating that biology, while being a useful inspiration to ANN
architecture is by far not the only means of advancement in AI.

In conclusion, the main advances driving the current success of ANNs are
focused on how ANNs are trained, not what exactly they are. These advances are
a mix of mainly pragmatic engineering choices and some questionable biological
inspiration which is also a good engineering choice.

##  4 Intelligence

Comparison of the machinery of ANNs with biological neural networks has so far
escaped simple conclusions. Here I will try to compare the intelligence (high-
level functionality) that arises from artificial and biological neural
networks. To obtain truly conclusive evidence of a deeper relationship between
ANNs and biological intelligence it would be useful to see biological-like
behaviour emerge from a system that we have not explicitly designed to do so.
Using emergent properties of simple neural networks to explain little
understood properties of the brain, such as the idiosyncratic functional
divisions between brain areas, was the agenda with which I first entered
academic research myself.

An early motivation for research into ANNs, whilst they were not yet powerful
enough to address practical tasks, was the similarity in their performance to
some aspects of human behaviour. David Rumelhart and Jay McClelland started an
entire research program within the area of psychology by exploiting these
similarities. They demonstrated that when reading partially obscured words,
ANNs and humans infer the obscured parts in a strikingly similar fashion [14].
Later on, Matt Lambon-Ralph demonstrated that when exposed to a new language,
ANNs pick-up words in a similar way to human children [4]. There are many
other examples demonstrating similarities between very simple ANNs and human
behaviour.

The similarity between the performance of simple ANNs and humans is not very
impressive given that these particular networks cannot manage to solve
anything beyond toy tasks, and are therefore very far from modelling anything
like human intelligence. Recently, however, ANNs gained the ability to perform
some basic real-world tasks. Research on these demonstrated similarities in
the internal state of a Deep Convolutional Neural Network and the human brain
when processing the same visual stimuli [9]. Specifically, it has been
demonstrated that the representations in both systems are based on
hierarchical pattern recognition.

These results should, however, surprise no one. It has been known to
neuroscientists for a long time that the visual processing in the brain is
akin to hierarchical pattern recognition. The ANNs of Guclu and Gerven are
only a mathematical specification of a hierarchical pattern recognition
system. The similarity therefore reflects a design decision, not an emergent
property, and can be hardly used to argue deeper similarities between ANNs and
biological intelligence. In fact, most of the emergent properties make a
convincing argument against functional similarities between ANNs and
biological intelligence.

It has been long argued that the amount of data required to train a
contemporary ANN is in a clear juxtaposition to animals being able to learn
from very few observations. In 2017 Google Deep Mind published a paper in
Nature boasting that they managed to train a network called AlphaGoZero within
only 36 hours, that not only beats any human Go player, but also AlphaGo – the
previous strongest Go player [18]. A deeper dive into the aforementioned
Nature paper, however, reveals that in those 36 hours AlphaGoZero played 4.9
million games. This roughly corresponds to 559 years of non-stop Go at human
speed, further emphasising how distant are the data requirements of ANNs are
from humans learning the same task.

A similar issue was well-documented by Dubey and colleagues in 2018 [3]. ANNs
were found to need much prolonged training to learn the basics of playing
Montezuma’s Revenge (a vintage video game) compared to people. After mixing-up
the textures in the game (so that, for example, the sky had a ground texture)
the game stopped making sense for humans while the ANNs learned the altered
game in approximately same amount of time as the original. This finding is
suggestive of a significant difference between the way ANNs and humans tackle
novel situations. Humans apply their past knowledge to novel situations – for
example they may attempt to climb a ladder rendered in a game, despite the
immense visual difference from a real-world ladder. However, this makes it
very difficult for humans to make sense of a sky that looks like a ground. The
ANNs took longer to learn the game in the first place because they are unable
to apply knowledge obtained in different contexts (e.g. real-life), and they
were also uninhibited by prior assumptions when learning something new. It is
far from clear how to make ANNs apply knowledge gathered in different
contexts444There has been some progress in the field of Transfer Learning
which aims at enabling ANNs to perform these tasks, however even the most
advanced systems are profoundly limited to specific tasks/contexts..

This problem – lack of ability to perform transitive inference - is in my view
the largest barrier for advancement of AI. Hector Levesque succinctly
summarises the problem in his book titled “Common Sense, the Turing Test, and
the Quest for Real AI” by asking a question: How would crocodile do in a
steeplechase? While finding an answer may seem trivial to you, it requires the
amount of transitive inference as of yet never seen in any ANN, bar those
specifically crafted for this purpose - which are unable to perform other
functions to a reasonable standard.

Even if ANNs could perform transitive inference, their prior knowledge would
still be fundamentally different from ours, probably leading to profoundly
different behaviour. This is rooted in the fact that the training sets
utilized by ANNs are so different to those available to animals. Linda B.
Smith has been spearheading a collection of headcam footage from toddlers [1],
which is as far from ML object recognition datasets as it could be. The most
prevalent objects are faces, and somewhat abstract imagery such as wall
corners or branches pictured against the sky; a stark difference to the
general cross-section of images from the internet used to train ANNs. I see a
significant potential in ANNs trained on this footage to perform in a more
similar way to humans than any previous models.

Transitive inference might simply be a functionality that we haven’t yet
achieved. However, there are good reasons to believe that even the current
success of ANNs in particular tasks is achieved in a way very different to
humans. Consider the two attempts (shown in figure 3) I made to estimate my
own demographics from an image, by a demographics-estimating product offered
by the company Clarifai. The first attempt was fairly accurate, concluding the
picture 99% likely to be a man, 51% likely to be age 27 and 98% likely to be
of white “multicultural appearance”. The second attempt resulted in 84% likely
to be a woman, 61% likely to be age 23 (thanks Clarifai!) and 40% likely to be
of hispanic “multicultural appearance”. I was also judged to be 18% likely to
be black or African American.

|  |
---|---|---
99% man |  | 84% woman
51% age 27 |  | 61% age 23
98% white |  | 40% hispanic, 18% black
Figure 3: Two attempts to judge author’s demographics made by a company
offering demographics-estimation by ANNs (Clarifai) from a photograph. The
shocking difference in the estimated demographics suggests that the features
used are fundamentally different from those used by humans for the same task.

The argument that I’d like to make here is not that Clarifai’s demographics-
estimating product is bad (I leave making that judgement to the reader), but
rather that the principles on which it estimates the demographics must be
fundamentally different to how humans accomplish the task. Sure, people have
varying ability to estimate the demographics; however, if someone estimates
the demographics from the first picture as well as Clarifai did, it seems
highly unlikely they would fail so miserably on the second picture.

The stark difference in features used for visual object recognition between
ANNs and humans is perhaps best illustrated by adversarial examples [8]. These
examples are tiny perturbations made to an original image that trick an ANN
into identifying an object as something entirely different to what it did
before the perturbation. Adversarial examples won’t trick humans as we can
mostly not even detect the perturbation. It turns out the same is true the
other way around; ANNs are not tricked by human visual illusions [20].

The intelligence displayed by ANNs can be very similar to that of humans as
long as the ANN was successfully trained for that purpose. For instance an ANN
trained to identify images will be good at that task, just like biological
intelligence is, but it will hardly replicate the way how we accomplish that
task as that was not the objective of the training. Therefore the performance
on a task the ANN was trained for can never suggest similarity in the basis of
intelligence, but rather success in the engineering task. Furthermore, ANNs
require orders of magnitude more data to learn from than humans, and there is
no known way how to make them successfully perform transitive inference yet –
a task we perform nearly all the time to our great advantage.

##  5 Conclusion

This article has compared artificial and biological Intelligence from a number
of angles. I proposed a view that there is no reason to expect different
computational properties from two different universal function approximators –
which both artificial and biological neurons undoubtedly are. Therefore, there
is a decent reason not to worry about the impact of huge structural
differences between biological and artificial neurons on the intelligent
behaviour that emerges from this machinery. What seems more important is how
learning of the desired function is performed. There is very little known
resemblance between artificial and biological intelligence in this respect,
except that both are capable of succesfully learning many of the tasks we are
interested in. The lack of resemblance between these two types of intelligence
is highlighted by the huge differences in performance on any task the ANNs
were explicitly trained to mimick biological intelligence.

A significant proportion of the widely circulated criticism relating to lack
of biological plausibility of the ANN machinery seems to stem from a lack of
up-to-date knowledge of neurobiology. However, the difference between the two
types of neurons remains very significant even after doing away with broken
arguments, especially when we consider the perplexing complexity revealed by
contemporary neurobiology. The difference is exacerbated by the fact that,
contrary to how they are presented in popular knowledge, neurons are not the
only cells capable of computation and communication. Such capabilities likely
spread beyond the nervous system or brain, and are mostly unknown.
Furthermore, currently even the properties of neurons are poorly understood by
science. Focusing on the veracity of the representation of biological neurons
therefore seems rather naive.

The story gets little more complicated when we look at the structure of ANNs
beyond individual neurons. Some of the technological advances that are driving
the current ML revolution are clearly biologically implausible (identity
matrix weight initialisation etc.), and at the same time are crucial in
allowing ANNs to achieve their current performance. On the other hand,
implementation of some ideas that are biologically well-grounded, such as
hierarchical pattern recognition, have made a crucial and positive
contribution to the performance of ANNs.

In the last section I discussed the functional differences between
intelligence of ANNs and animals. There has been a long-standing research
effort demonstrating similar patterns of performance between trivial ANNs and
humans. However, it is unclear what the performance of toy models on toy tasks
can tell us about true intelligence.

Recently efforts have been made to identify common patterns between ANNs that
can actually solve real-world tasks, and the human brain. In my view these
findings have so far failed to identify any patterns of similarity that are
not simple design decisions made during construction of the ANN. To the
contrary, there is a plenty of evidence suggesting the two types of
intelligence are fundamentally different. The amount of data needed for
learning differs by several orders of magnitude, and there is no capability to
use previous knowledge or perform transitive reasoning. Finally, the features
exploited to achieve a task are fundamentally different.

I would like to end this article on a less damning note. Despite the profound
differences in both structure and function between ANNs and biological
intelligence it is necessary to remember that some of the fundamental
principles (parallel, hierarchical, non-linear etc.) of biological
intelligence applied in the ANNs led to their astonishing success, and there
might be more to follow.

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# BSM global fits with GAMBIT:
a Dark Matter EFT fit

Tomás E. Gonzalo  on behalf of the GAMBIT Community  Institute for Theoretical
Particle Physics and Cosmology (TTK),
RWTH Aachen University, D-52056 Aachen, Germany

###### Abstract

In this conference paper I present the first full global fit of a dark matter
effective field theory with the global fitting framework GAMBIT. I show the
results of exhaustive parameter space explorations of the effective dark
matter model, including a general set of operators up to dimension 7, and
using the most up-to-date constraints from direct and indirect detection of
dark matter, relic abundance requirements and collider searches for dark
matter candidates.

##  1 Introduction

The nature of dark matter (DM) and its interactions has been an extensively
explored subject for a long time. Due to the large landscape of plausible
models of DM, it is convenient to parametrise the DM interactions with
Standard Model (SM) particles in a model-agnostic way, using the Effective
Field Theory (EFT) approach [2]. EFT models often have vast parameter spaces
and thus smart sampling strategies are needed to fully explore them [3]. One
of the most powerful tools for such study is the GAMBIT framework [4]. GAMBIT,
the Global And Modular BSM Inference Tool, is an open-source global fitting
software able to perform statistical fits on a variety of BSM models [5, 6, 7,
8, 9], and thus it is perfectly suited for this purpose.

##  2 Dark Matter Effective Field Theory

The most generic interaction lagrangian for an effective field theory of DM
can be written as
$\mathcal{L}_{\rm{int}}=\sum_{a,d}(\mathcal{C}_{a}^{(d)}/{\rm\Lambda}^{d-4})\mathcal{Q}_{a}^{(d)}$,
where $\mathcal{Q}_{a}^{(d)}$ is the interaction operator, $d$ its dimension,
$\mathcal{C}_{a}^{(d)}$ the dimensionless Wilson coefficient (WC), and
${\rm\Lambda}$ is the scale of new physics. In this study we focus exclusively
on the interactions between DM particles ($\chi$) and quarks and gluons, as
they are the most relevant interactions for searches of DM. Hence, the
effective operators we consider are

$\scriptsize{\begin{array}[]{llllll}\mathcal{Q}_{1,q}^{(6)}&=(\overline{\chi}\gamma_{\mu}\chi)(\overline{q}\gamma^{\mu}q)\,,&\mathcal{Q}_{1}^{(7)}&=\frac{\alpha_{s}}{12\pi}(\overline{\chi}\chi)G^{a\mu\nu}G^{a}_{\mu\nu}\,,&\mathcal{Q}_{5,q}^{(7)}&=m_{q}(\overline{\chi}\chi)(\overline{q}q)\,,\\\
\mathcal{Q}_{2,q}^{(6)}&=(\overline{\chi}\gamma_{\mu}\gamma_{5}\chi)(\overline{q}\gamma^{\mu}q)\,,&\mathcal{Q}_{2}^{(7)}&=\frac{\alpha_{s}}{12\pi}(\overline{\chi}i\gamma_{5}\chi)G^{a\mu\nu}G^{a}_{\mu\nu}\,,&\mathcal{Q}_{6,q}^{(7)}&=m_{q}(\overline{\chi}i\gamma_{5}\chi)(\overline{q}q)\,,\\\
\mathcal{Q}_{3,q}^{(6)}&=(\overline{\chi}\gamma_{\mu}\chi)(\overline{q}\gamma^{\mu}\gamma_{5}q)\,,&\mathcal{Q}_{3}^{(7)}&=\frac{\alpha_{s}}{8\pi}(\overline{\chi}\chi)G^{a\mu\nu}\widetilde{G}^{a}_{\mu\nu}\,,&\mathcal{Q}_{7,q}^{(7)}&=m_{q}(\overline{\chi}\chi)(\overline{q}i\gamma_{5}q)\,,\\\
\mathcal{Q}_{4,q}^{(6)}&=(\overline{\chi}\gamma_{\mu}\gamma_{5}\chi)(\overline{q}\gamma^{\mu}\gamma_{5}q)\,,&\mathcal{Q}_{4}^{(7)}&=\frac{\alpha_{s}}{8\pi}(\overline{\chi}i\gamma_{5}\chi)G^{a\mu\nu}\widetilde{G}^{a}_{\mu\nu}\,,&\mathcal{Q}_{8,q}^{(7)}&=m_{q}(\overline{\chi}i\gamma_{5}\chi)(\overline{q}i\gamma_{5}q)\,,\\\
&&&&\mathcal{Q}_{9,q}^{(7)}&=m_{q}(\overline{\chi}\sigma^{\mu\nu}\chi)(\overline{q}\sigma_{\mu\nu}q)\,,\\\
&&&&\mathcal{Q}_{10,q}^{(7)}&=m_{q}(\overline{\chi}i\sigma^{\mu\nu}\gamma_{5}\chi)(\overline{q}\sigma_{\mu\nu}q)\,.\\\
\end{array}}$

Many of the constraints, such as direct detection, are computed using non-
relativistic operators, at lower scales. The running of the relativistic
operators above from the input scale (${\rm\Lambda}$), with mixing and
threshold effects included, as well as the matching to the non-relativistic
operators, is performed per parameter point by DirectDM v2.2.0 [10].
Throughout this study we consider $\mathcal{C}_{a}^{(d)}$ and ${\rm\Lambda}$
as independent parameters. This allows the study to use weaker bounds but with
larger range of validity in ${\rm\Lambda}$, as opposed to naive approaches
that only vary $C/\Lambda^{d-4}$. Therefore, the free parameters in this model
are the 6- and 7-dimensional WCs, $\mathcal{C}_{a}^{(6)}$ and
$\mathcal{C}_{a}^{(7)}$, the new physics scale ${\rm\Lambda}$, and the DM mass
$m_{\chi}$. In addition, a set of nuisance parameters are varied
simultaneously corresponding to the DM halo profile, SM masses and nuclear
parameters, amounting to a total of 24 scan parameters.

##  3 Likelihoods and Constraints

The effective operators described above predict potentially strong
interactions between DM and the particles in the SM. Hence, the parameters of
the EFT model, namely the WCs, the DM mass, $m_{\chi}$ and new physics scale
${\rm\Lambda}$, are strongly constrained by the various searches for DM. The
specific set of constraints employed in this study is as follows:

Direct detection

DM particles from the Galactic halo can scatter off nuclei. Various direct
detection experiments with ultra-pure targets can detect such rare processes,
and thus enforce strong constraints on the interaction cross-section of DM
particles. The direct detection experiments considered in this study are
CDMSlite, CRESST-II and -III, DarkSide 50, LUX 2016, PICO-60, PandaX 2016 and
2017, and XENON1T. We use DirectDM to compute the value of the non-
relativistic WCs, including all running and mixing effects, and DDCalc v2.2.0
[11] to calculate the likelihood for each of the relevant direct detection
experiments.

Relic abundance

The precise measurement of the relic abundance of DM by Planck, $\Omega_{\rm
DM}h^{2}=0.120\pm 0.001$, severely constrains the annihilation cross-section
of DM particles in the Early Universe. As a conservative approach, we assume
that the DM particles studied here do not constitute all of the observed DM,
and thus we allow parameter combinations were DM is underabundant, with a DM
fraction $f_{\chi}$. We use this DM fraction to scale the direct and indirect
detection signals. We compute the relic density (RD) using DarkSUSY v6.2.2
[12], from tree-level cross-sections calculated with CalcHEP v3.6.27 [13],
from the interactions generated with GUM [14]. In order to preserve EFT
validity, we ignore parameter points with ${\rm\Lambda}\leq 2m_{\chi}$.

Indirect detection

The annihilation of DM particles in regions of high density may produce
visible signals in various forms. These may be $\gamma$-rays from dwarf
spheroidal galaxies, collected by Fermi-LAT, or high energy neutrinos from the
annihilation of DM captured in the Sun, observed by IceCube. Furthermore, DM
annihilations in the Early Universe inject energy into the primordial plasma
and affect the reionisation history, which can be observed in the CMB by
Planck as changes in the optical depth. From the annihilation cross-section
computed by CalcHEP, the indirect detection likelihoods are computed for
$\gamma$-rays by gamLike v.1.0.1 [11], for neutrinos by Capt’n General v2.1
[15] and nulike v1.0.9 [16], and the constraints from the CMB by a combination
of DarkSUSY and DarkAges [17] via CosmoBit [18]. As before, meaningful DM
annihilations require ${\rm\Lambda}>2m_{\chi}$.

Collider physics

DM particles can be produced at LHC by proton-proton collisions. In this study
we focus on a $36\,\rm{fb}^{-1}$ CMS and a $139\,\rm{fb}^{-1}$ ATLAS search,
where a jet is produced via initial state radiation, leading to final states
with a single jet and missing transverse energy. Collider simulations can be
very CPU expensive, so we opted for pre-generating cross-section and
efficiency tables, from which we interpolated the yields. We generate Monte
Carlo events using MadGraph_aMC@NLO v2.6.6 (v2.9.2) [19] for the CMS (ATLAS)
analysis, which we later shower and hadronise using Pythia v8.1 [20]. To
ensure EFT validity in the collider searches we take two approaches. One where
the $\not{E}_{T}$ spectrum has a hard cut-off at ${\rm\Lambda}$, and another
where the spectrum has a smooth drop-off with varying slope,
$(\not{E}_{T}/{\rm\Lambda})^{-a}$. The value of $a$ depends on the specific UV
completion, but since we are performing an EFT analysis, we vary $a$ as a free
parameter of the model.

##  4 Results

We present the results of this study in two parts. First we show the results
with a “capped” LHC likelihood, i.e. where the model fits the data equally or
worse than the background hypothesis. And second, we allow the full LHC
likelihood, where we will use both of the EFT approaches described earlier,
with hard and smooth cut-offs on the $\not{E}_{T}$ spectrum.

Figure 1: Profile likelihood in the full range (left) and most interesting
region (centre) of the $m_{\chi}$–$\Lambda$ plane with dimension-6 operators,
a capped LHC likelihood and the relic density as an upper bound. The right-
hand plot shows the low mass region with saturated relic density. The best-fit
point is indicated by the white star, the white lines indicate the 68% and 95%
confidence level regions, and the grey region is excluded by EFT validity.

Figure 1 (left and centre) shows the profile likelihood in the $m_{\chi}$ vs
${\rm\Lambda}$ plane for the $d=6$ operators. The most interesting features of
these figures are the large parameter region excluded by EFT validity
(${\rm\Lambda}>2m_{\chi}$), the upper limit on ${\rm\Lambda}$ at high masses
due to the RD requirement, and the strongly constrained region for
${\rm\Lambda}>200$ GeV and $m_{\chi}<200$ GeV where the LHC searches are the
dominant constraint. The best fit corresponds to a slight excess on the Fermi-
LAT data.

It is worth exploring the scenario where the relic abundance is exactly
saturated ($f_{\chi}\approx 1$). This can be seen in the right panel of Figure
1. It is now not possible to saturate the RD bound for low mases
$m_{\chi}\lesssim 100$ GeV, as it is incompatible with gamma ray and CMB
constraints. Because of the shrunken parameter region, the best fit point with
saturated relic density predicts up to $10$ signal events at the next
generation of direct detection experiments, e.g. LZ.

Figure 2: Profile likelihood in the $m_{\chi}$–${\rm\Lambda}$ parameter plane
with only $d=6$ operators (left, centre) and all $d=6$ and $d=7$ operators
(right), as well as the full LHC likelihood with a hard (left) and smooth
(centre, right) cut-off in the $\not{E}_{T}$ spectrum. Stars, lines are shaded
regions are as in Figure 1.

If the LHC likelihood is included in full, it becomes the dominant constraint
for $m_{\chi}<500$ GeV, as can be seen in Figure 2. In the $d=6$ case, there
is clearly a preference now for high ${\rm\Lambda}$ values, following the bin-
wise excesses in the CMS and ATLAS analyses. When using a hard cut-off on the
$\not{E}_{T}$ spectrum (left), the profile likelihood shows various 1$\sigma$
regions, corresponding to values of ${\rm\Lambda}$ where the signal-to-
background is maximized individually for the CMS (${\rm\Lambda}\approx 700$
GeV) and ATLAS (${\rm\Lambda}\gtrsim 1$ TeV) analyses. With a smooth cut-off
(centre), the best fit is a combination of all excesses and thus the fit is
slightly better. Although a promising feature, this slight preference over the
background may be an artefact of the EFT construction, and realistic UV
completions may not be able to fit individual bin excesses as the EFT approach
does.

Lastly, one can study the effect of adding the dimension-7 operators. Overall
the addition of the $d=7$ operators does not noticeably increase the allowed
parameter space. The most notable difference is that for low $m_{\chi}$ it is
possible to saturate the RD and avoid LHC constraints, which was impossible
with only $d=6$ operators. Furthermore, with the full LHC likelihoods it is
now possible to simultaneously fit the LHC excesses at high masses and the
Fermi-LAT excess at low masses, due to the increased parameter volume, as seen
in the right panel of Figure 2.

##  5 Conclusions

I have presented in this conference article a summary of the first global
analysis of a DM effective field theory with the full set of operators up to
dimension 7. We find that there are large regions of the parameter space
allowed where the scattering and annihilation cross-sections are suppresed, so
that it is possible to evade direct and indirect detection constraints, while
producing the right amount of relic density. We have found that constraints
from LHC searches are strong for low DM masses and high new-physics scales,
and that it is possible to find a slight preference for a DM signal.
Nevertheless, the log likelihood ratio between the various best-fit points and
the background-only hypothesis is always small, so we do not find a
significant preference for a DM signal in any of our scans. This work opens
the door for many subsequent studies, e.g. with lepton operators, specific UV
completions or non-trivial flavour structures, among others.

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