corresponding NLO predictions more prone to perturbative instabilities. The
GIVS results for the NLO decay widths are again similar to the PRTS results,
which are included in the detailed study of NLO predictions for
$\text{H}\to\text{W}\text{W}/\text{Z}\text{Z}\to 4f$ based on various types of
renormalization schemes presented in Ref. Denner:2018opp . For $\text{H}\to
4f$, predictions, thus, should be based on the OS-type or symmetry-inspired
schemes proposed in Ref. Denner:2018opp .

##  4 Conclusions

Applying $\overline{\text{MS}}$ renormalization conditions to mass parameters
or mixing angles leads to subtle issues in the precise definition of vev
parameters if EW corrections are included in predictions. Technically, these
issues concern the treatment of tadpoles in the course of renormalization. In
the past, mostly two tadpole schemes have been in use, called PRTS and FJTS in
this paper, which, however, both have benefits and drawbacks. The PRTS is
based on the idea to renormalize parameters in such a way that a field is
expanded about the minimum of the effective Higgs potential. For Higgs fields
developing vevs this means that no potentially large corrections to the vevs
enter predictions, but gauge-dependent tadpole loop contributions enter the
relations between bare parameters of the theory (at least in the usually
adopted Higgs representations), leading to gauge-dependent parametrizations of
observables. The alternative scheme, called FJTS, avoids gauge dependences by
just redistributing tadpole contributions in predictions via field shifts
without altering parameter relations, but on cost of an expansion about a
field configuration that corresponds to the vacuum only in lowest order. By
experience this procedure is prone to artificially large corrections which
jeopardize the stability of perturbation theory.

To improve on this unsatisfactory situation, in a preceding paper
Dittmaier:2022maf  we have proposed a hybrid scheme for treating tadpole
contributions, dubbed Gauge-Invariant Vacuum expectation value Scheme (GIVS),
which unifies the benefits and avoids the shortcomings of the PRTS and FJTS.
Perturbative stability is achieved by an expansion about the true vacuum field
configuration, and gauge dependences are avoided by employing non-linear Higgs
representations, in which CP-even Higgs fields and their vevs are gauge-
invariant quantities. As an application in the SM, we had shown in Ref.
Dittmaier:2022maf  that the conversion of OS-renormalized to
$\overline{\text{MS}}$-renormalized masses involves only very moderate EW
corrections, in contrast to the situation in the FJTS. In this paper, we have
applied the GIVS to the $\overline{\text{MS}}$ renormalization of Higgs mixing
angles in two scalar extensions of the SM: a Higgs singlet extension (called
SESM), and the Two-Higgs-Doublet model (THDM), the renormalization of which
has been extensively discussed in the literature in recent years.

To apply the GIVS to the SESM and THDM, we have first formulated the Higgs
sectors in a non-linear way. While the SESM requires only minor modifications
beyond the SM formulation, the generalization to the THDM is quite non-trivial
and interesting in its own right. We have described the transition from
linearly to non-linearly realized Higgs doublets in detail and have presented
the (non-polynomial) kinetic Lagrangians of the THDM Higgs field in a way that
renders the generation of all Feynman rules (which involve arbitrarily many
Goldstone legs) quite simple. While kinetic Lagrangians become more
complicated in non-linear Higgs representations, the Higgs potentials become
simpler, because all would-be Goldstone fields drop out there by construction.

To illustrate the perturbative stability of the results based on
$\overline{\text{MS}}$ renormalization with the GIVS tadpole treatment, we
have discussed the NLO predictions for the important four-body decays of CP-
even neutral Higgs bosons,
$\text{h}/\text{H}\to\text{W}\text{W}/\text{Z}\text{Z}\to 4\,$fermions based
on $\overline{\text{MS}}$-renormalized Higgs mixing angles. In all cases, we
have found that the (gauge-independent) GIVS results are very close to the
corresponding (gauge-dependent) PRTS results, thus, sharing the superiority
over the FJTS results w.r.t. perturbative stability. The new GIVS variants of
the $\overline{\text{MS}}$ schemes will be made available in a forthcoming
public version of the Monte Carlo program Prophecy4f for the considered four-
body Higgs decays.

Finally, in the THDM we have made the observation that the GIVS variant for an
$\overline{\text{MS}}$-renormalized parameter $\tan\beta=v_{2}/v_{1}$ produces
exactly the same one-loop renormalization constants for $\tan\beta$ as the
PRTS variant in the class of $R_{\xi}$ gauges, i.e. that effectively the
$\overline{\text{MS}}$ PRTS renormalization for $\tan\beta$ does not suffer
from gauge dependences after reinterpretation as GIVS results. This
observation is particularly interesting in the context of EW renormalization
of supersymmetric models, in which $\tan\beta$ is often
$\overline{\text{MS}}$-renormalized with PRTS-like tadpole prescriptions. We
conjecture that this interesting fact generalizes to higher orders as well
and, thus, can put higher-order predictions in supersymmetric models obtained
with the PRTS in $R_{\xi}$ gauges on a gauge-invariant basis.

### Acknowledgements

Ansgar Denner is gratefully acknowledged for helpful discussions and for
comments on the manuscript. Moreover, the authors acknowledge support by the
state of Baden-Württemberg through bwHPC, by the DFG through grant no INST
39/963-1 FUGG, and by the BMBF under contract 05H21VFCA1. H.R.’s research is
funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation)—project no. 442089526; 442089660.

## Appendix

##  Appendix A Parameter conversion tables

The full set of input parameters of the various SESM and THDM scenarios, as
used in Sections 3.1.3 and 3.2.3, can be found in Refs. Altenkamp:2018bcs  and
Altenkamp:2017ldc ; Altenkamp:2017kxk , respectively. Here we list results for
the conversion of those non-standard input parameters that are subject to
different renormalization schemes for the SESM and THDM. The input parameters
are defined in the on-shell input schemes OS and OS12101010Although NLO
results in the OS and OS12 schemes are completely independent of the tadpole
treatment, there is some slight dependence on the tadpole scheme (beyond NLO
accuracy) used in these schemes if input parameter conversions between schemes
are done in a self-consistent way i.e. not truncated at the NLO level (see
Ref. Denner:2019vbn  for more details). For completeness, we mention that we
apply the PRTS variant in the OS and OS12 schemes. in the SESM and THDM,
respectively, and subsequently converted into the various
$\overline{\text{MS}}$ renormalization schemes. Details on the conversion
procedure can also be found in Sect. 3.5 of Ref. Denner:2018opp .

###  A.1 SESM scenarios

Table 4 shows the results for the (full) conversion of parameters from the OS
renormalization scheme used as input scheme to $\overline{\text{MS}}$ schemes
for the considered benchmark scenarios.

BHM200+ BHM200- BHM400 BHM600 Ren. scheme tadpoles $s_{\alpha}$ $\lambda_{12}$
$s_{\alpha}$ $\lambda_{12}$ $s_{\alpha}$ $\lambda_{12}$ $s_{\alpha}$
$\lambda_{12}$ OS $0.29$ $0.07$ $-0.29$ $-0.07$ $0.26$ $0.17$ $0.22$ $0.23$
$\overline{\text{MS}}$ FJTS $0.321$ $0.077$ $-0.316$ $-0.076$ $0.264$ $0.173$
$0.212$ $0.222$ $\overline{\text{MS}}$ PRTS $0.302$ $0.073$ $-0.304$ $-0.073$
$0.267$ $0.175$ $0.226$ $0.236$ $\overline{\text{MS}}$ GIVS $0.301$ $0.073$
$-0.302$ $-0.073$ $0.266$ $0.174$ $0.225$ $0.236$

Table 4: Conversion of parameters from the OS scheme as input scheme to the
different $\overline{\text{MS}}$ renormalization schemes at the central scale
$\mu_{0}=M_{\text{h}}$ performed in the PRTS, using the bare parameters for
$\tan\alpha$ and $\lambda_{12}$ in the matching.

This table completes Tab. 14 of Ref. Denner:2018opp , where the input
conversion to other schemes can be found. We recall that the included
renormalization schemes employ $\lambda_{1}$ (not $\lambda_{12}$) as
independent running $\overline{\text{MS}}$-renormalized parameter, although
the scenarios are fixed by the values of $\lambda_{12}$.

The moderate corrections in the input parameter conversions reflect the
perturbative stability of the $\overline{\text{MS}}$ schemes with all the
considered tadpole treatments in all considered scenarios. Note also that the
corrections are almost identical for the PRTS and GIVS variants, as expected
from the small differences in the tadpole terms of the renormalization
constant $\delta\alpha$ of the mixing angle $\alpha$.

###  A.2 THDM scenarios

Table 5 shows the corresponding conversion in the THDM from the OS12
renormalization scheme used as input scheme to the different
$\overline{\text{MS}}$ renormalization schemes.

A1 A2 B1 B2 Ren. scheme tadpoles $c_{\alpha\beta}$ $t_{\beta}$
$c_{\alpha\beta}$ $t_{\beta}$ $c_{\alpha\beta}$ $t_{\beta}$ $c_{\alpha\beta}$
$t_{\beta}$ OS12 ($\alpha,\beta$) $0.1$ $2.0$ $0.2$ $2.0$ $0.15$ $4.5$ $0.3$
$3.0$ $\overline{\text{MS}}$ ($\alpha,\beta$) FJTS $0.090$ $1.93$ $0.157$
$1.91$ $0.061$ $3.82$ $-0.031$ $3.70$ $\overline{\text{MS}}$ ($\alpha,\beta$)
PRTS $0.137$ $1.90$ $0.225$ $1.91$ $0.153$ $4.40$ $0.176$ $2.83$
$\overline{\text{MS}}$ ($\alpha,\beta$) GIVS $0.135$ $1.90$ $0.222$ $1.91$
$0.151$ $4.40$ $0.174$ $2.83$ $\overline{\text{MS}}$ ($\lambda_{3},\beta$)
FJTS $0.128$ $1.92$ $0.238$ $1.88$ $0.403$ $2.04$ $-0.037$ $3.62$
$\overline{\text{MS}}$ ($\lambda_{3},\beta$) PRTS/GIVS $0.132$ $1.90$ $0.232$
$1.91$ $0.160$ $4.40$ $0.190$ $2.83$

Table 5: Conversion of parameters from the OS12 scheme as input scheme to the
different $\overline{\text{MS}}$ renormalization schemes at the central scale
$\mu_{0}=(M_{\text{h}}+M_{\text{H}}+{M}_{A_{0}}+2{M}_{\text{H}^{\pm}})/5$
performed in the PRTS, using the bare parameters for $\tan\alpha$ and
$\tan\beta$ in the matching.

This table completes Tab. 15 of Ref. Denner:2018opp , where the input
conversion to other schemes can be found. As compared to the SESM, the
conversion effects for the considered THDM scenarios are more pronounced. The
conversion to $\overline{\text{MS}}$(FJTS) becomes perturbatively unstable
(and fails completely) for scenarios B1 and B2. On the other hand, the PRTS
and GIVS tadpole treatments lead to moderate corrections, reflecting
perturbative stability. As expected, the results of the PRTS and GIVS variants
almost agree for the $\overline{\text{MS}}$ ($\alpha,\beta$) scheme; in the
$\overline{\text{MS}}$ ($\lambda_{3},\beta$) scheme these two tadpole variants
produce identical results as explained in Section 3.2.1.

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# A Novel Blind Source Separation Framework Towards Maximum Signal-To-
Interference Ratio

Jianjun Gu, Longbiao Cheng, Dingding Yao, Junfeng Li, and Yonghong Yan This
research is partially supported by the National Key Research and Development
Program of China (No. 2020YFC2004104) and the National Natural Science
Foundation of China (Nos. 12104483, 11911540067).All the authors are with the
Key Laboratory of Speech Acoustics and Content Understanding, Institute of
Acoustics, Chinese Academy of Sciences, and University of Chinese Academy of
Sciences, Beijing 100190, China. (e-mail<EMAIL_ADDRESS>
###### Abstract

This letter proposes a new blind source separation (BSS) framework termed
minimum variance independent component analysis (MVICA), which can potentially
achieve the maximum output signal-to-interference ratio (SIR) while also
allowing more flexibility in real implementations. The statistical
independence assumption has been the foundation of the most dominant BSS
techniques in recent decades. However, this assumption does not always hold
true and the accurate probabilistic modeling of source is inherently
difficult. To overcome these limitations and improve the separation
performance, the MVICA framework is rigorously derived by optimizing the
design of these independence-based BSS algorithms with the maximum SIR
criterion. A deep neural network-supported implementation of MVICA is
subsequently described. Experimental results under various conditions show the
superiority of MVICA over the state-of-the-art BSS algorithms, in terms of not
only SIR but also signal-to-distortion ratio and automatic speech recognition
rate.

###### Index Terms:

Blind source separation, independent component analysis, maximum SIR, deep
neural network.

##  I Introduction

Blind source separation (BSS) technology [1] aims at determining the original
source signals from the observed mixtures with minimal prior knowledge about
the mixing process, which has found prominent applications in speech
communication and speech-based human-machine interaction systems [2]. For BSS
in the (over)determined case (number of microphones $\geq$ number of sources),
statistical independence between the source signals is widely assumed to
estimate the demixing matrix [3, 4], bringing out a family of independence-
based BSS algorithms. Among them, frequency domain independent component
analysis (FDICA) [5, 6] and independent vector analysis (IVA) [7, 8, 9] are
the most classical ones. The auxiliary function technique is then introduced
to more stably and efficiently update the demixing matrices for ICA (AuxICA)
[10] and IVA (AuxIVA) [11, 12]. The separation performance of these methods
highly relies on the quality of the source model, which thus attracts wide
research interests. The independent low-rank matrix analysis (ILRMA) algorithm
[13] incorporate a nonnegative matrix factorization (NMF) decomposition-based
source model into IVA, leading to significant improvements in music source
separation. Given the powerful modeling capability of deep neural network
(DNN), DNN is recently utilized to model the spectral pattern of signals and
achieves the state-of-the-art (SOTA) BSS performance [14, 15, 16, 17, 18].

Despite the foregoing improvements, there is still a problem that the maximum
independence criterion does not always lead to a satisfactory separation
result. Because the independence assumption does not always hold true [19, 20]
and the precise modeling of source remains a challenging problem [17]. As a
supplement to the maximum independence criterion, additional penalties such as
the geometrical constraints [21, 22, 23] and the sparsity-based constraints
[24, 25], have been demonstrated to improve the performance when incorporated
into these BSS methods. However, these penalties rely on prior knowledge of
sources [25]. By contrast, the maximum signal-to-interference ratio (SIR)
criterion [26] directly optimizes the separation performance with respect to
interference suppression and target source preservation, and can be generally
used for any sources. Unfortunately, how to incorporate the maximum SIR
criterion into these BSS methods remains a problem.

In this letter, our goal is to maximize the output SIR for each source while
preserving the advantage of fast and stable solutions from the mainstream
independence-based BSS algorithms. To this end, we follows the auxiliary
function technique-based update rules [10, 12] in these algorithms, but
propose to optimize the choice of the weighted covariance matrix toward
maximizing the output SIR. The optimal weighted covariance matrix is proved to
be the interference covariance matrix, form which the maximum-SIR demixing
matrix can be directly computed, thus also avoiding the difficulty of source
modeling. As a result, a new BSS framework called minimum variance independent
component analysis (MVICA) is obtained. Any methods for estimating the
interference covariance matrix can be incorporated into MVICA, therefore
giving more freedom to actual applications and further studies. A DNN-
supported implementation of MVICA is finally presented. Experiments reveal
that MVICA significantly outperforms the SOTA BSS algorithms, in terms of SIR,
signal-to-distortion ratio (SDR) and automatic speech recognition (ASR) rate.

##  II Problem Formulation

We address the determined BSS problem where $K$ sources are observed by $K$
microphones ($K\geq 2$). The short-time Fourier Transform (STFT) coefficients
of the source, the observed, and the estimated signals are denoted as
$s_{k}(f,t)$, $x_{k}(f,t)$ and $y_{k}(f,t)$, respectively, where
$t=1,...,N;f=1,...,N_{f};$ and $k=1,...,K$ index the time frames, frequency
bins and sources, respectively. For the reverberant mixtures, when the window
size used in the STFT is sufficiently long compared with the impulse responses
between sources and microphones, the mixing process can be modeled by

$\bm{x}(f,t)={A}(f)\bm{s}(f,t),$ (1)

where
$\bm{d}(f,t)=[d_{1}(f,t),d_{2}(f,t),\cdots,d_{K}(f,t)]^{T}\in\mathbb{C}^{K}(d\in\\{s,x,y\\})$
denotes the vector representation of signal, and

${A}(f)=[\bm{a}_{1}(f),\bm{a}_{2}(f),\cdots,\bm{a}_{K}(f)]\in\mathbb{C}^{K\times
K},$ (2)

is the mixing matrix containing the acoustic transfer functions from the
source positions to the microphones, and the superscript T denotes the
transpose.

The primary aim of BSS is to estimate a demixing matrix

$W(f)=[\bm{w}_{1}(f),\bm{w}_{2}(f),\cdots,\bm{w}_{K}(f)]^{H}\in\mathbb{C}^{K\times
K},$ (3)

so that the source signals can be recovered by

$\bm{y}(f,t)={W}(f)\bm{x}(f,t),$ (4)

and the superscript H denotes the Hermitian transpose.

##  III Conventional Method

The conventional independence-based BSS algorithms normally estimate the
demixing matrix $W(f)$ as the one that maximizes measure related to
independence of estimated sources. Let $p\left(\bm{y}_{k}(t)\right)$ be the
probabilistic model, many of them fall into a optimization problem of the
following form [27]:

$\mathop{\arg\min}_{\left[W(f)\right]_{f=1}^{N_{f}}}\sum_{k,t}G\left\\{\bm{y}_{k}(t)\right\\}-\sum_{f}\log|\operatorname{det}W(f)|,$
(5)

where $G\left\\{\bm{y}_{k}(t)\right\\}=-\log p\left(\bm{y}_{k}(t)\right)$ is
the contrast function, $\operatorname{det}(\cdot)$ denote the determinant
operator and

$\bm{y}_{k}(t)=[y_{k}(1,t),...,y_{k}(N_{f},t)]^{T}\in\mathbb{C}^{N_{f}}.$ (6)

For example, FDICA, IVA and ILRMA can be obtained by assuming the following
source models [27], respectively:

FDICA $\displaystyle\rightarrow
p\left(\bm{y}_{k}(t)\right)\propto\prod_{f}\exp\left\\{-\left|y_{k}(f,t)\right|\right\\},$
(7) IVA $\displaystyle\rightarrow
p\left(\bm{y}_{k}(t)\right)\propto\exp\left\\{-\left\|\bm{y}_{k}(f,t)\right\|_{2}\right\\},$
ILRMA $\displaystyle\rightarrow
p\left(\bm{y}_{k}(t)\right)\propto\prod_{f}\exp\left\\{-\frac{\left|y_{k}(f,t)\right|^{2}}{\tilde{y}_{k}(f,t)}\right\\},$

where $\tilde{y}_{k}(f,t)$ denotes the source variance estimated by NMF and
$\|\cdot\|_{2}$ denotes $L_{2}$ norm of a vector.

To solve Eq.(5) based on the auxiliary function technique [12], the weighted
covariance matrix is further introduced as

$V_{k}(f)=E\left\\{\frac{G^{{}^{\prime}}\left(\bm{y}_{k}(t)\right)}{r_{k}(f,t)}\bm{x}(f,t)\bm{x}^{H}(f,t)\right\\},$
(8)

where $G^{{}^{\prime}}$ denotes the differential of $G$, $E\\{\cdot\\}$ denote
expectation over frames and $r_{k}(f,t)$ gets assigned the value of
$|y_{k}(f,t)|$, $\left\|\bm{y}_{k}(f,t)\right\|_{2}$ and
$|\tilde{y}_{k}(f,t)|$ for FDICA, IVA and ILRMA, respectively. Fast and stable
demixing matrix update rules can then be obtained as

$\bm{w}_{k}(f)\leftarrow V_{k}^{-1}(f)W^{-1}(f)\bm{e}_{k},$ (9)
$\bm{w}_{k}(f)\leftarrow\bm{w}_{k}(f)/\sqrt{\bm{w}^{H}_{k}(f)V_{k}(f)\bm{w}_{k}(f)},$
(10)

where $\bm{e}_{k}$ denotes the unit vector with the $k$th element unity. In
practice, Eqs.(4), (8), (9) and (10) are applied in order for all $k$ and $f$.
The iteration is repeated until convergence is obtained.

##  IV Proposed Method

###  IV-A Derivation of The MVICA Algorithm Framework

Our goal is to determine the maximum-SIR demixing matrix with the above update
rules, where $\bm{w}_{k}(f)$ can be directly estimated from $V_{k}(f)$ by
substituting Eq.(9) into (10) as

$\bm{w}_{k}(f)\leftarrow\frac{V_{k}^{-1}(f)\tilde{\bm{a}}_{k}(f)}{\sqrt{\tilde{\bm{a}}_{k}(f)V_{k}^{-1}(f))\tilde{\bm{a}}_{k}(f)}},$
(11) $\tilde{\bm{a}}_{k}(f)=\tilde{W}^{-1}(f)\bm{e}_{k}.$ (12)

$\tilde{W}(f)$ in Eq.(12) is the estimated demixing matrix obtained in the
previous iteration. So we first determine the optimal $V_{k}(f)$, from which
the maximum-SIR demixing matrix can then be computed using Eqs.(9) and (10).
$V_{k}(f)$ depends on the source models as indicated in Eq.(8), but we avoid
the difficulty of source modeling and directly derive its optimal form, since
a explicit source model is not always necessary [25]. The optimization problem
is thus formulated as

${V}_{k}^{\text{opt}}(f)=\mathop{\arg\max}\limits_{V_{k}(f)}\text{SIR}_{k}(f)$
(13)

where the output SIR is defined as

$\text{SIR}_{k}(f)=\frac{\bm{w}_{k}^{H}(f)\bm{\Phi}_{\bm{S}_{k}}(f)\bm{w}_{k}(f)}{\bm{w}_{k}^{H}(f)\bm{\Phi}_{\bm{N}_{k}}(f)\bm{w}_{k}(f)}$
(14)

and $\bm{\Phi}_{\bm{S}_{k}}(f)$ and $\bm{\Phi}_{\bm{N}_{k}}(f)$ denote the
covariance matrix of the $k$th source image and the corresponding
interference, respectively, i.e.,

$\bm{\Phi}_{\bm{S}_{k}}(f)=E\left\\{\bm{S}_{k}(f,t)\bm{S}_{k}^{H}(f,t)\right\\}\in\mathbb{C}^{K\times
K},$ (15)
$\bm{\Phi}_{\bm{N}_{k}}(f)=E\left\\{\bm{N}_{k}(f,t)\bm{N}_{k}^{H}(f,t)\right\\}+\mathcal{E}*I\in\mathbb{C}^{K\times
K},$ (16) $\bm{S}_{k}(f,t)=\bm{a}_{k}(f)s_{k}(f,t),$ (17)
$\bm{N}_{k}(f,t)=\bm{x}(f,t)-\bm{S}_{k}(f,t),$ (18)

where a diagonal loading term $\mathcal{E}*I$ ($\mathcal{E}$ is a tiny
constant and $I$ denotes the identity matrix) is introduced into
$\bm{\Phi}_{\bm{N}_{k}}(f)$ to stabilize the matrix inverse operation, it is
thus positive definite and can decomposed as

$\bm{\Phi}_{\bm{N}_{k}}(f)=\bm{P}^{H}_{k}(f)\bm{P}_{k}(f),$ (19)

where $\bm{P}_{k}(f)$ is the $K\times K$ invertible matrix. Substituting
Eq.(17) into (15), $\bm{\Phi}_{\bm{S}_{k}}(f)$ is a rank-1 matrix given by

$\bm{\Phi}_{\bm{S}_{k}}(f)=E\left\\{\left|s_{k}(f,t)\right|^{2}\right\\}\bm{a}_{k}(f)\bm{a}_{k}^{H}(f).$
(20)

Substituting Eqs.(19), (20) and (11) into (13), the optimization problem can
then be rewritten as

$\mathop{\arg\max}\limits_{V_{k}(f)}\frac{E\left\\{\left|s_{k}(f,t)\right|^{2}\right\\}\bm{D}^{H}\bm{P}^{-H}_{k}(f)\bm{a}_{k}(f)\bm{a}_{k}^{H}(f)\bm{P}^{-1}_{k}(f)\bm{D}}{\bm{D}^{H}\bm{D}},$
(21)

where

$\bm{D}=\bm{P}_{k}(f)V_{k}^{-1}(f)\tilde{\bm{a}}_{k}(f).$ (22)

According to the Cauchy-Schwarz inequality [28], we have

$\displaystyle\bm{D}^{H}$
$\displaystyle\bm{P}^{-H}_{k}(f)\bm{a}_{k}(f)\bm{a}_{k}^{H}(f)\bm{P}^{-1}_{k}(f)\bm{D}$
(23) $\displaystyle\leq$
$\displaystyle\left\\{\bm{D}^{H}\bm{D}\right\\}\left\\{\bm{a}_{k}^{H}(f)\bm{P}^{-1}_{k}(f)\bm{P}^{-H}_{k}(f)\bm{a}_{k}(f)\right\\}.$

Thus, the output SIR satisfies

$\text{SIR}_{k}(f)\leq
E\left\\{\left|s_{k}(f,t)\right|^{2}\right\\}\bm{a}_{k}^{H}(f)\bm{\Phi}_{\bm{N}_{k}}^{-1}(f)\bm{a}_{k}(f)$
(24)

and the equality sign is satisfied if and only if

$\lambda\bm{P}^{-H}_{k}(f)\bm{a}_{k}(f)=\bm{D},$ (25)

that is,

$\lambda\bm{\Phi}_{\bm{N}_{k}}^{-1}(f)\bm{a}_{k}(f)=V_{k}^{-1}(f)\tilde{\bm{a}}_{k}(f),$
(26)

where $\lambda$ is an positive constant, and $\tilde{\bm{a}}_{k}(f)$ defined
in Eq.(12) is the approximation of $\bm{a}_{k}(f)$ since the desired demixing
matrix should satisfy $\tilde{W}(f)\approx A^{-1}(f)$. Therefore, the weighted
covariance matrix that is capable of the maximum SIR is exactly the
interference covariance matrix, i.e.,

${V}_{k}^{\text{opt}}(f)=\frac{1}{\lambda}\bm{\Phi}_{\bm{N}_{k}}(f),$ (27)

where the scale parameter $1/\lambda$ can be omitted because of the
normalization operation followed in Eq.(10).

By replacing Eq.(8) with Eq.(27) for the conventional independence-based BSS
algorithm, we obtain the MVICA algorithm framework. As summarized in Algorithm
1, MVICA is compose of two discrete steps: estimating the interference
covariance matrix for each source, which in turn is used to blindly compute
the demixing matrix $\bm{w}_{k}(f)$ based on the update rules given in Eq.(9).
It is worth noting that MVICA omits the normalization operation in Eq.(10).
This computational simplification comes because the scale ambiguity for the
separated signals will be resolved by the post processing.

###  IV-B Differences Between MVICA And The Existing Methods

As given in Eq.(24), the maximum SIR obtained by MCIVA is equivalent to that
of the minimum variance distortionless response (MVDR) beamformer [30, 31, 29]
and the generalized eigenvalue (GEV) beamformer [26]. Moreover, both MVICA and
the GEV beamformer strive to maximize the out SIR. However, different from the
update rules adopted by MVICA, the GEV beamformer and its extension in [36]
estimate the filter coefficients via GEV decomposition, and we will compare
their performances in the experiment part. MVICA is also preferred over the
MVDR beamformer and its DNN-supported variants [32, 33, 34], because it avoids
the extra estimation of the relative transfer function (RTF) [32] or the
covariance matrix of sources [34], which is a remarkable benefit.

###  IV-C DNN-based Implementation of MVICA

The key problem now is the accurate estimation of the interference covariance
matrix $\bm{\Phi}_{\bm{N}_{k}}(f)$. Any methods for estimating
$\bm{\Phi}_{\bm{N}_{k}}(f)$ can be incorporated into MVICA. Inspired by
success of the mask-based beamformers [32, 33, 34], we introduce a DNN-
supported implementation of MVICA, which estimates the interference signal by

$\tilde{\bm{N}}_{k}(f,t)=\bm{m}_{k}(f,t)\bm{x}(f,t),$ (28)

where $\bm{m}_{k}\in\mathbb{C}^{N_{f}\times T}$ is the complex-valued mask
computed by the DNN and its weights are shared across channels. The
interference covariance matrix can then be computed as:

$\tilde{\bm{\Phi}}_{\bm{N}_{k}}(f)=\frac{\sum_{t=1}^{T}\tilde{\bm{N}}_{k}(f,t)\tilde{\bm{N}}_{k}^{H}(f,t)}{\sum_{t=1}^{T}\bm{m}_{k}^{H}(f,t)\bm{m}_{k}(f,t)}+\mathcal{E}*I.$
(29)

Algorithm 1 The MVICA algorithm framework

1\. Estimate the interference covariance matrix
$\tilde{\bm{\Phi}}_{\bm{N}_{k}}(f)$ for all $k$.

2\. Compute the demixing matrix for all $f$:

for $l$ of number of iterstions $L$ do

for $k=1,2,...,K$ do
$\bm{w}_{k}(f)\leftarrow\tilde{\bm{\Phi}}_{\bm{N}_{k}}^{-1}(f)W^{-1}(f)\bm{e}_{k}.$
(30)

end for

end for

For the DNN architecture, the deep complex convolutional recurrent network
(DCCRN) [35] is adopted given its outstanding performance in speech
enhancement. The multi-channel complex mixture spectrograms are stacked as
input features to estimate $\bm{m}_{k}$ for all $k$ simultaneously. For the
loss function, we use the complex-domain scale-invariant signal-to-distortion
ratio in [37] combined with the spectral mean square error loss to measure the
estimation error of $\tilde{\bm{N}}_{k}(f,t)$. Permutation invariant training
[38] is also utilized because of the ambiguity of the output order. Unlike the
previous DNN-supported methods [14, 15, 16, 18], which update the source model
parameters with DNN iteratively, leading to the expensive computational cost
and the mismatch between training and testing data, the proposed MVICA
algorithm learn the mask directly from the mixtures signals.

##  V Experimental Results

TABLE I: Separation performances of the different methods on the test mixing data ($K$: the number of sources). $\bm{K}$ |  | Methods |  | SIR Improvements (dB) | SDR Improvements (dB) |  | WER(%) | CER(%)
---|---|---|---|---|---|---|---|---
|  | 100ms | 200ms | 300ms | 400ms | 100ms | 200ms | 300ms | 400ms |  | 100ms | 200ms | 100ms | 200ms
2 |  | AuxIVA [12] |  | 11.28 | 9.91 | 8.71 | 7.56 | 7.44 | 5.51 | 3.95 | 2.90 |  | 48.56 | 60.52 | 32.31 | 39.04
| ILRMA [13] |  | 16.54 | 12.03 | 10.50 | 8.83 | 11.90 | 7.02 | 4.79 | 3.27 |  | 39.53 | 59.08 | 24.90 | 39.73
| Kang [17] |  | 21.46 | 15.72 | 13.90 | 11.44 | 15.58 | 9.51 | 6.69 | 4.65 |  | 17.03 | 25.95 | 9.20 | 14.61
| IDLMA [14] |  | 20.90 | 16.14 | 14.04 | 11.89 | 16.19 | 10.13 | 6.94 | 5.01 |  | 20.83 | 42.46 | 12.89 | 27.26
| MVICA-DNN |  | 25.92 | 20.54 | 17.70 | 15.22 | 18.65 | 11.63 | 7.52 | 5.61 |  | 8.36 | 16.97 | 3.77 | 7.70
3 |  | AuxIVA [12] |  | 9.12 | 7.13 | 6.45 | 5.46 | 5.72 | 3.95 | 3.13 | 2.21 |  | 93.57 | 115.60 | 52.78 | 70.19
| ILRMA [13] |  | 12.02 | 8.57 | 7.03 | 5.41 | 8.56 | 5.14 | 3.53 | 2.08 |  | 76.36 | 112.07 | 41.19 | 66.88
| Kang [17] |  | 13.59 | 11.29 | 9.32 | 10.11 | 9.83 | 7.14 | 5.15 | 5.20 |  | 51.39 | 64.42 | 25.48 | 34.98
| IDLMA [14] |  | 17.87 | 13.51 | 9.99 | 12.26 | 13.10 | 9.06 | 5.85 | 6.57 |  | 23.83 | 36.61 | 11.21 | 20.18
| MVICA-DNN |  | 19.16 | 16.39 | 13.73 | 13.74 | 13.36 | 10.37 | 7.60 | 6.80 |  | 16.47 | 32.58 | 7.26 | 16.70

###  V-A Data Preparation and Experimental Setup

We train our models on synthesized convolutive mixtures of speech. 14,400
speech samples are selected from the WSJ0 dataset [39] in total. While, 150
room impulse responses (RIRs) are generated for each of the 2-source and
3-source case using the image source method [40] in a simulated 4 m $\times$ 4
m $\times$ 3 m room. The reverberation time is chosen randomly in the 100 ms
to 400 ms interval. The microphone spacing is 2 cm and the arrangement of
sources is selected at random. All the models are trained for 20 epochs, and
100,000 mixtures are generated in each epoch by convolving the sources with
the RIRs randomly selected from the training set. Another 800 and 1,600
mixtures are generated with unseen sources and RIRs for validation and test,
respectively. All the signals are sampled at 16 kHz. The STFT frame size is
4096 with half overlap. To eliminate scaling ambiguity of the separated
signals, the estimated demixing matrix is rescaled based on the minimal
distortion principle [41]. The iteration number of the demixing matrix updates
is set to $L=5$.

Besides the conventional methods, another two SOTA DNN-supported BSS
algorithms are adopted for comparison (independent deeply learned matrix
analysis (IDLMA) [14] and the Kang method in [17]). To make a fair comparison,
IDLMA and the Kang method use the same DCCRN architecture to directly learn
their target from mixtures. The performances of all the methods are evaluated
by the SIR and SDR calculated by the BSS toolbox [42], and also word error
rate (WER) and character error rate (CER) of an ASR system trained using the
WSJ0 dataset [39] with the ESPnet framework [43].

Figure 1: Oracle performances of different methods under various conditions.

###  V-B Results and Discussion

We begin by comparing the oracle separation performances that can be
potentially achieved by different methods. More specifically, the target
spectrogram and frame-level variance of each source are directly given to
AuxICA/ILRMA/IDLMA and the AuxIVA/Kang method, respectively. The proposed
MVICA algorithm and the GEV beamformer (with blind analytical normalization
[26]) are also applied with the ideal interference covariance matrix. As Fig.
1 shows, MVICA has the best oracle performance in terms of SDR and SIR,
demonstrating its superiority over the existing BSS algorithms. In addition,
MVICA yields obviously superior performance over the GEV beamformer especially
in terms of SDR scores, although they both aim to maximize the output SIR.
Table 1 further shows the practical performances on the test set. The DNN-
supported MVICA algorithm significantly outperforms the other methods with
respect to SIR, SDR, WER and CER. The effectiveness of DNN-based interference
covariance matrix estimation for MVICA is thus confirmed. It is worth
mentioning that, while MVICA is derived to maximize SIR, the other metrics
have seen significant gains as well. As an example, Fig. 2 displays the source
spectrograms predicted by different methods. It can be clearly seen that MVICA
can recover the source signal with less residual interference.

Figure 2: The spectrograms of a 2-source mixture (a), and the sources
estimated by the Kang method (c), IDLMA (d) and the DNN-supported MVICA
algorithm (e), where the red box marks the residual interference components.

##  VI Conclusion

In this letter, a new determined BSS algorithm framework called MVICA is
mathematically derived towards maximizing the output SIR, where the optimal
weighted covariance matrix is proved to be the interference covariance matrix.
MVICA avoids the difficulty of source modeling and achieves the SOTA BSS
performances. It is also superior to the MVDR and GEV beamforming techniques
because it avoids the extra estimation of RTF and introduces fewer signal
distortions. In the future work, we intend to investigate more efficient ways
to estimate the interference covariance matrix and apply the proposed methods
to the noisy environments.

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