音響システム特論第2回 advanced topics of acoustic systems day 2 · 2019-09-30 ·...

音響システム特論第2回Advanced Topics of Acoustic Systems Day 2

小山翔一 / Shoichi Koyama

東京大学大学院情報理工学系研究科Graduate School of Information Science and Technology, The University of Tokyo

Oct. 1, 2019

小山翔一 / Shoichi Koyama (UTokyo) Acoustic Systems Day2 Oct. 1, 2019 1 / 63

講義概要 / Summary

目的 / Goals• 音響信号を対象としたモデリングや最適化の方法論を学ぶことで，信号処理における発展的な数理的手法を，応用を通して理解することを目指す。

• Gain understanding of advanced mathematical techniques insignal processing through applications by learning modelingand optimization methods for acoustics signals.

概要 / Summary• 前半は，音響信号を対象とした統計的推定法やアレイ信号処理，逆問題に関して講義する。具体的には，指向性制御，音源位置推定，空間音響，音場計測・制御などである。後半は，より発展的な内容に関して，ゲスト講師による講義を行う。

• The first half is lectures about statistical estimation, array signalprocessing, and inverse problems for acoustic signals. Morespecifically, beamforming, source localization, spatial audio,and sound field analysis and control. The second half is talks byguest lecturers about more advanced topics.


講義概要 / Summary

キーワード / Keywords• 音響信号処理,統計的信号処理,アレイ信号処理,スパースモデリング,逆問題,指向性制御,音源位置推定,空間音響,音場計測,音場制御

• Acoustic signal processing, Statistical signal processing, Arraysignal processing, Sparse modeling, Inverse problems,Beamforming, Source localization, Spatial audio, Sound fieldanalysis, Sound field control

講義は日本語で行いますが，スライドはできるだけ英語にします。 /Lectures are in Japanese, but slides are in English as much aspossible.


講義日程 / Schedule

Aセメスター,火曜 2限@セミナー室 BSemester A, Tuesday 2nd period @Seminar room Bスケジュール / Schedule9/24 統計的推定法の復習 / Statistical Estimation Revisited10/1 スパースモデリング (1) / Sparse Modeling (1)10/8 スパースモデリング (2) / Sparse Modeling (2)10/15 アレイ信号処理 / Array Signal Processing10/29 休講 / Canceled11/5 物理音響学の復習 / Physical Acoustics Revisited11/12 空間音響の基礎 / Basics of Spatial Audio11/19 音場の計測と制御 / Sound Field Analysis and Control11/2612/312/1012/171/7

ゲスト講師による講演 / Talks by Guest Lecturers


講義日程 / Schedule

ゲスト講師による講演 / Talks by Guest Lecturers11/26 山岸昌夫先生 (東京工業大学) / Prof. Masao Yamagishi

(Tokyo Tech)“凸最適化と音響システム (仮) / Convex Optimization andAcoustic Systems”

12/3 高道慎之介先生 (東京大学) / Prof. Shinnosuke Takamichi(UTokyo)“音声合成・変換の基礎 (仮) / Basics of Speech Synthesis andConversion”

12/10 郡山知樹先生 (東京大学) / Prof. Tomoki Koriyama (UTokyo)“ガウス過程回帰による音声・音響モデリング (仮) / Speechand Acoustic Modeling Based on Gaussian Process”

12/17 小泉悠馬氏 (NTT研究所) / Dr. Yuma Koizumi (NTT Lab.)“実世界における音響信号処理と機械学習（仮） / AcousticSignal Processing and Machine Learning in Real World”

1/7 猿渡洋先生 (東京大学) / Prof. Hiroshi Saruwatari (UTokyo)“教師無し最適化に基づく音響信号処理 / Acoustic Signal Pro-cessing Based on Unsupervised Learning”


講義資料と成績評価 / Materials and Grades

講義資料 / Course materials• Downloadable at http://www.sh01.org/ja/teaching/

ソースコード / Codes• Several codes related to this course are available athttps://github.com/sh01k/teaching

成績評価 / Grading system• 出席 / Attendance• レポート課題 (2回) / Reporting assignments (twice)


http://www.sh01.org/ja/teaching/

https://github.com/sh01k/teaching

Table of Contents

1 Sparse modeling (1)What is sparse modeling?Sparsity-inducing normGreedy algorithmConvex relaxation


Table of Contents



What is sparse modeling?

Representing data with small number of nonzero coefficientsof basis functions.Such a representation enables to reconstruct signals fromsmall number observations.



Representing images by wavelet bases.

(a) Original (b) 10% of coefs (c) 3% of coefs



Distribution of coefficients of wavelet bases is sparse.

0 1 2 3 4 5 6 7

Index of basis functions 104

0

2

4

6

8

Am

pli

tude

of

coef

fici

ents

By decomposing image with wavelet bases, most ofcoefficients can be ignored.



Consider that data y ∈ RM is represented by N dictionaryvectors (N > M ).

y =

N∑n=1

xndn

This redundant dictionary leads to underdetermined linearequation as

y = Dx,

whereD = [d1, . . . ,dN ] ∈ RM×N (dictionary matrix) andx = [x1, . . . , xN ] ∈ RN .



Basic problem of sparse modeling is to find the sparsestsolution from countless possible solutions.


Table of Contents



Vector norm

Sparsity of the solution is induced by norm. Definition ofvector norm is revisited here.The most well-known norm will be ℓ2-norm or Euclidean norm∥x∥2.

∥x∥2 :=√

x21 + x22 + . . .+ x2N =√⟨x,x⟩,

where ⟨·, ·⟩ represents ℓ2-inner product or Euclidean innerproduct.


Vector norm

DefinitionIf non-negative real value ∥x∥ satisfies the following conditions forall x ∈ RN , ∥x∥ is referred to as norm of x.1 ∥αx∥ = |α|∥x∥ for any α ∈ R2 ∥x+ y∥ ≤ ∥x∥+ ∥y∥3 ∥x∥ = 0 ⇔ x = 0

Several norms can be defined for RN . ℓp-norm (1 ≤ p < ∞) is ageneralization of ℓ2-norm defined as

∥x∥p :=

(N∑

n=1

|xn|p) 1

p


Vector norm

ℓp-norm corresponds to ℓ2-norm for p = 2.Another important norm is ℓ1-norm, which is summation ofthe absolute values of the elements.

∥x∥1 =N∑

n=1

|xn|

By taking the limit of p → ∞, ℓ∞-norm or maximum-norm isdefined as

∥x∥∞ := maxn∈{1,...,N}

|xn|


Vector normIn the context of sparse modeling, the definition of norm issometimes extended to 0 ≤ p < 1.ℓp-pseudo norm is defined for 0 < p < 1 as

∥x∥p :=

(N∑

n=1

|xn|p) 1

p

For p = 0, ℓ0-pseudo norm or cardinality is defined as thenumber of nonzero elements.

∥x∥0 := limp→0

N∑n=1

|xn|p = |supp(x)|,

where

supp(x) := {n ∈ {1, 2, . . . , N} | xn ̸= 0} .

Hereafter, ∥x∥p for 0 ≤ p < 1 is also referred to as norm.小山翔一 / Shoichi Koyama (UTokyo) Acoustic Systems Day2 Oct. 1, 2019 18 / 63

Minimum-norm solution

Underdetermined linear equation is typically solved asminimum-norm solution.

minimizex∈RN

1

2∥x∥22 subject to y = Dx

This optimization has closed-form solution obtained by usingLagrange’s method of underdetermined multipliers as

x̂ = DT(DDT)−1y

Obviously, sparsity of the solution is not induced by ℓ2-norm.


ℓ0 optimization

The sparsest solution of the underdetermined linear equationy = Dx will be obtained by solving the following optimizationproblem:

minimizex∈RN

∥x∥0 subject to y = Dx

Unfortunately, this problem is NP-hard, which is classified ascombinatorial optimization.


Sparse optimization

How to efficiently obtain sparse solution of underdeterminedlinear equation?Several approaches to sparse optimization:

• Greedy algorithm• Convex relaxation• Majorization-minimization algorithm• Probabilistic inference


Table of Contents



Greedy algorithm

Greedy algorithm is efficient strategy to directly obtain localoptimal solution of ℓ0-optimization.

minimizex∈RN

∥x∥0 subject to y = Dx

First, mutual coherence of dictionary matrixD = [d1, . . . ,dN ] isdefined as

µ(D) = maxi,j∈{1,...,N},i ̸=j

|⟨di,dj⟩|∥di∥2∥dj∥2

The maximum value of µ(D) is 1 because |⟨x,y⟩| ≤ ∥x∥2∥y∥2for ∀x,y ∈ RM from Cauchy–Schwarz inequality. The minimumvalue is 0.


Greedy algorithm

TheoremIf linear equation y = Dx has a solution x obeying

1

2

(1 +

1

µ(D)

)> ∥x∥0,

this solution is necessarily the sparsest possible. [Elad 2010]

Assume that K ∈ N satisfying µ(D) < 1/2K − 1 exists andy = Dx has solution satisfying ∥x∥0 = K, i.e., K-sparsesolution. According to the above theorem, this solution is thesparsest solution.If K = 1, one can obtain such a solution by finding dn parallelto y. It can be achieved by finding n that minimizes error e(n)defined as

e(n) := minimizex∈R

∥xdn − y∥22 = ∥y∥22 −⟨dn,y⟩2

∥dn∥22小山翔一 / Shoichi Koyama (UTokyo) Acoustic Systems Day2 Oct. 1, 2019 24 / 63

Greedy algorithm

If K-sparse solution is necessary to obtain for general K, onemust find K out of N dictionary vectors. In greedy algorithm,however, solution of local problem is sequentially selected.Global optimal solution cannot be always obtained, butgreedy algorithm is much faster than exhaustive search.Various algorithms...

• (Orthogonal) matching pursuit• Iterative hard-thresholding algorithm• Compressive sampling matching pursuit (CoSaMP)


Orthogonal matching pursuit

In orthogonal matching pursuit (OMP), minimization problemfor 1-sparse vector is sequentially solved K times.1 Find 1-sparse vector x(1) that minimizes ∥Dx− y∥22. Index of

nonzero element n̂ is set as support S(1) = {n̂}.2 Repeat for k = 2, . . . ,K to find 1-sparse vector x(k) that

minimizes ∥Dx− r(k−1)∥22 where r(k−1) is residual at kth step.Index of nonzero element is added to supportS(k) = S(k−1) ∪ {n̂}.


Orthogonal matching pursuitInitialize x(0) = 0, r(0) = y −Dx(0), S(0) = ∅for k = 1, . . . ,K doSweep:

ϵn = minimizexn

∥xndn − r(k−1)∥22 = ∥r(k−1)∥22 −⟨dn, r

(k−1)⟩2

∥dn∥22Update support:

n̂ = arg minn/∈S(k−1)

ϵn, S(k) = S(k−1) ∪ {n̂}

Update provisional solution:

x̂(k) = arg minxS(k)

∥y −DS(k)xS(k)∥22 =(DT

S(k)DS(k)

)−1DT

S(k)y

Update residual:r(k) = y −DS(k)xS(k)

end for小山翔一 / Shoichi Koyama (UTokyo) Acoustic Systems Day2 Oct. 1, 2019 27 / 63

Table of Contents



Convex relaxation

ℓ0-optimization is hard to solve. Another norm can also lead tosparse solution?

Figure: Contour of ∥x∥p = c with constant c in 2D case (replace qwith p). [Bishop 2006]


Convex relaxationℓp-norm of 0 < p ≤ 1 can also lead to sparse solution.

Figure: Contour ∥x∥p = c and constraint y = Dx in 3D case. [Elad 2010]


Convex relaxation

Consider ℓ1-optimization problem:

minimizex∈RN

∥x∥1 subject to Dx = y

Find a point on (hyper) planeDx = y such that ℓ1-norm isminimized.In many cases, intersection of ∥x∥1 = c andDx = y is atcoordinate axis. This point includes 0 in some coordinates.


Convex relaxation

Benefit of the use of ℓ1-norm is that the objective function∥x∥1 becomes convex.Method using ℓ1-norm optimization is referred to as basispursuit.

Figure: Plot of |x|p. [Elad 2010]


Convex relaxation

Sparse solution is also induced by regularization term ofℓ1-norm:

minimizex∈RN

1

2∥y −Dx∥22 + λ∥x∥1

This problem is referred to as ℓ1 regularization or leastabsolute shrinkage and selection operator (LASSO).Under some conditions, solution of ℓ1-optimization problemcorresponds to that of ℓ0-optimization problem.


Basis pursuitBasis pursuit solves ℓ1-optimization problem:

minimizex∈RN


This problem can be reformulated as linear programmingproblem (線形計画問題).

• x is represented as x = u− v. Here, u has positive element of xand the rest is 0. v has absolute value of negative element of xand the rest is 0.

• Vector z is defined as z := [uT,vT]T ∈ R2N .• Then, ∥x∥1 is rewritten as ∥x∥1 = 1T(u+ v), where 1 is all 1vector. Besides,Dx = D(u− v) = [D,−D]z.

• Thus, ℓ1 minimization problem is reformulated as linearprogramming problem as

minimizez

1Tz subject to y = [D,−D]z and z ⪰ 0

Standard convex optimization tools can be used.小山翔一 / Shoichi Koyama (UTokyo) Acoustic Systems Day2 Oct. 1, 2019 34 / 63

Basis pursuit

Easily implemented by using MATLAB...1 z = linprog(ones(2*N,1),[],[],[D,-D],

y,zeros(2*N,1),[]));2 x = z(1:N)-z(N+1:2*N);


Basis pursuit

Numerical example: y ∈ R30, x ∈ R60,D ∈ R30×60, and K = 2.

0 5 10 15 20 25 30

m

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

(a) Observation y

0 10 20 30 40 50 60

n

-0.5

0

0.5

1

1.5

(b) True x

0 10 20 30 40 50 60

n

-0.5

0

0.5

1

1.5

(c) Basis pursuit

0 10 20 30 40 50 60

n

-0.5

0

0.5

1

1.5

(d)Minimum norm


Convex optimization

General optimization tool is useful, but is not suitable for largeproblem because of its computational cost.Efficient algorithms based on convex optimization areintroduced.


Convex optimization: preliminaries

First, several definitions for convex optimization areintroduced.Set C is convex if for any x,y ∈ C and any λ ∈ [0, 1], we have

λx+ (1− λ)y ∈ C

Figure: Example of convex and nonconvex sets [Boyd+ 2004]


Convex optimization: preliminariesSuppose function f : RN → R ∪ {∞}.Effective domain (実効定義域) of function f is defined as set ofx such that f(x) has real value.

dom(f) :={x ∈ RN | f(x) < ∞

}If effective domain of f is not empty, i.e., at least one xsatisfying f(x) < ∞ exists, the function f is proper.As an example, consider the following function:

f(x) =

{0, ∥x∥2 ≤ 1

∞, ∥x∥2 > 1

Effective domain of this function is

dom(f) = {x ∈ RN | ∥x∥2 ≤ 1}

and f is proper.小山翔一 / Shoichi Koyama (UTokyo) Acoustic Systems Day2 Oct. 1, 2019 39 / 63


Function f : RN → R ∪ {∞} is convex if for all x,y ∈ dom(f)and λ ∈ [0, 1], we have

f(λx+ (1− λ)y) ≤ λf(x) + (1− λ)f(y)

Figure: Example of convex function [Boyd+ 2004]



Epigraph of function f : R → R ∪ {∞} is defined as

epi(f) :={(x, t) ∈ RN × R | x ∈ dom(f), f(x) ≤ t

}

Figure: Epigraph of function f [Boyd+ 2004]

If epgraph of function f , epi(f), is closed set, function f isclosed function.


Convex optimization: proximal operator

Proximal operator (近接作用素) proxγf is defined for closedproper convex function f : RN → R ∪ {∞} and real numberγ > 0 as

proxγf (v) := arg minx∈dom(f)

{f(x) +

1

2γ∥x− v∥22

}proxγf (v) is a point that compromises between minimizing fand being near to v (proximal point). The parameter γ is arelative weight between these terms.



Figure: Proximal operator. [Parikh+ 2013]

Contour of convex function with boundary of dom(f) isdepicted. Evaluating proxγf at blue points moves them to thecorresponding red points.



Proximal operator of f can be interpreted as a kind of gradientstep for f as in gradient method. Actually, under someconditions,

proxγf (v) ≈ v − γ∇f(v)

when γ is small and f is differentiable.Note that proximal operator can be defined for general closedproper convex function even if it is not differentiable.Effective domain of f , dom(f), is invariant set of proximaloperator of f , i.e., points in dom(f) is necessarily projected indom(f) by proxγf .Fixed points of proximal operator of f are the minimizers of f ,i.e., proxγf (x∗) = x∗ iff x∗ minimizes f .


Convex optimization: proximal algorithm

Proximal algorithm is an algorithm that minimizes convexfunction f by repeatedly applying proxγf to some initial point.

Proximal algorithmSet initial vector x(0) and positive sequence γ0, γ1, . . ., then repeat

x(k+1) = proxγkf (x(k)) k = 0, 1, . . .

Vector sequence generated by proximal algorithm, {x(k)},converges to one of solutions minimizing f if

∑∞k=0 γk = ∞

holds.Proximal algorithm work under fairly general conditions, e.g.,the case that the functions are nonsmooth. Besides, it can bevery fast because there are simple proximal operators formany types of functions.


Examples of proximal operators

Indicator function (指示関数) for close convex set C ∈ RN isdefined as

IC(x) :=

{0, x ∈ C∞, x /∈ C

IC(x) is closed proper convex function if C is closed convex set.Proximal operator for indicator function is represented as

proxIC(v) = arg minx∈RN

{IC(x) +

1

2∥x− v∥22

}= arg min

x∈C∥x− v∥22

:= ΠC(v)

proxIC is projection operator to C, ΠC .


Examples of proximal operators

Next, proximal operator of ℓ1-norm is considered.

f(x) = ∥x∥1 =N∑

n=1

|xn|

Proximal operator of ℓ1-norm is obtained as

proxγf (v) = arg minx∈RN

{∥x∥1 +

1

2γ∥x− v∥22

}

= arg minx∈RN

N∑n=1

{|xn|+

1

2γ(xn − vn)

2

}where vn is nth element of v.This problem is simplified to the minimization problem of eachelement.


Examples of proximal operatorsEach element of proxγf (v) is written as

[proxγf (v)

]n=

vn − γ, vn ≥ γ

0, −γ < v < γ

v + γ, v ≤ −γ

:= Tγ(vn)

Tγ is referred to as soft thresholding operator.For simplicity, denoted as proxγf (v) = Tγ(v).


Proximal algorithm for ℓ1-optimization

ℓ1-optimization problem:

minimizex∈RN


Set of x ∈ RN satisfying constraint conditionDx = y isdenoted as C.

C :={x ∈ RN | Dx = y

}This set is closed convex set in RN .By using indicator function IC(x), ℓ1-optimization problem canbe rewritten as

minimizex∈RN

∥x∥1 + IC(x)

∥x∥1 + IC(x) is closed proper convex function.



Consider to use proximal algorithm for the unconstrainedoptimization problem

f(x) := ∥x∥1 + IC(x)

Since this objective function has a form of f = f1 + f2, it isdifficult to directly apply proximal algorithm. Here,

f1(x) := ∥x∥1, f2(x) := IC(x)

Algorithm separately applying proximal operators of f1 and f2will be efficient. Such an algorithm is referred to as proximalsplitting algorithm.



Douglas–Rachford splitting, one of the proximal splittingalgorithms [Combettes+ 2011], is introduced here.Suppose the following optimization problem:

minimizex∈RN

f1(x) + f2(x)

Both f1 and f2 are closed proper convex function and theirproximal operators proxγf1 and proxγf2 are derived in closedform.

Douglas–Rachford algorithmSet initial value z(0) and parameter γ > 0, then repeat

x(k+1) = proxγf1(z(k))

z(k+1) = z(k) + proxγf2(2x(k+1) − z(k))− x(k+1)


Proximal algorithm for ℓ1-optimizationFor ℓ1-optimization problem, f1(x) = ∥x∥1 and f2(x) = IC(x),thus,

proxγf1(v) = Tγ(v), proxγf2(v) = ΠC(v)

Douglas–Rachford algorithm for ℓ1-optimizationSet initial value z(0) and parameter γ > 0, then repeat

x(k+1) = Tγ(z(k))

z(k+1) = z(k) +ΠC(2x(k+1) − z(k))− x(k+1)

ΠC is specifically obtained based on Lagrange’s method ofundetermined multipliers as

ΠC(v) = v +DT(DDT)−1(y −Dx)


Proximal algorithm for ℓ1-regularization

Consider ℓ1-regularization (or LASSO) problem:

minimizex∈RN

1

2∥Dx− y∥22 + λ∥x∥1

Douglas–Rachford algorithm can also be applied to thisproblem, but more efficient algorithm is introduced here,which can be applied if either term is differentiable.



Suppose the following optimization problem:

minimizex∈RN

f1(x) + f2(x)

f1 is differentiable convex function satisfying dom(f1) = RN

and f2 is closed proper convex function. f2 can beindifferentiable.This type of problem can be solved by proximal gradientmethod.

Proximal gradient algorithmSet initial value x(0) and positive parameter γ > 0, then repeat

x(k+1) = proxγf2(x(k) − γ∇f1(x

(k))), k = 0, 1, . . .



Each step of proximal gradient algorithm is written as

φ(x) = proxγf2(x− γ∇f1(x))

φ(x) is reformulated by using the definition of proximaloperator as

φ(x) = arg minz∈RN

{f2(z) +

1

2γ∥z − x+ γ∇f1(x)∥22

}= arg min

z∈RN

{f1(x) +∇f1(x)

T(z − x) +1

2γ∥z − x∥22︸︷︷︸

:=f̃1(z,x)

+f2(z)}

f̃1(z,x) is second-order approximation of f1(z) around x ∈ RN ;therefore, f̃1(z,x) ≥ f1(x) and equality holds for z = x.


Proximal algorithm for ℓ1-regularizationIn proximal gradient algorithm, x is iteratively updated as

x(k+1) = arg minx

{f̃1(x,x

(k)) + f2(x)}

f̃1(x,x(k)) + f2(x) is upper bound on f1(x) + f2(x) when

γ ∈ (0, 1/L], where L is Lipschitz constant of ∇f1.Proximal gradient algorithm can be regarded as a kind ofmajorization-minimization algorithm.



Function f is Lipschitz continuous if a constant L > 0 existssuch that

∥f(x)− f(y)∥2 ≤ L∥x− y∥2holds for any x,y ∈ RN . Minimum of such constant L isreferred to as Lipschitz constant.When γ ≤ 1/L, vector sequence obtained by proximal gradientalgorithm {x(k)} converges to the optimal solution x∗, and

∥x(k+1) − x∗∥2 ≤ ∥x(k) − x∗∥2, k = 0, 1, . . .

holds. Besides, by denoting f(x) = f1(x) + f2(x),

f(x(k))− f(x∗) ≤ L∥x(0) − x∗∥222k

, k = 0, 1, . . .

holds. Therefore, its convergent rate is O(1/k).



Proximal gradient algorithm is applied to ℓ1-regularizationproblem.

f1(x) =1

2∥Dx− y∥22, f2(x) = λ∥x∥1

Gradient of f1(x) is

∇f1(x) = DT(Dx− y)

Proximal operator of f2(x) is

proxγλ(v) = Tγλ(v)



Proximal gradient algorithm for ℓ1-regularization problemSet initial value x(0) and parameter γ > 0, then repeat

x(k+1) = Tγλ(x(k) − γDT(Dx(k) − y)

), k = 0, 1, . . .

This algorithm is referred to as iterative shrinkagethresholding algorithm (ISTA).Lipschitz constant L of ∇f1 is L = σmax(D

TD), whereσmax(D

TD)means maximum of absolute value of eigenvalueofDTD (spectral radius). Therefore, γ should be set as

γ ≤ 1

σmax(DTD)



Fast ISTA (FISTA) accelerates convergence of ISTA by using(k − 1)th-step value x(k−1) to calculate x(k+1) in addition to x(k)

(Nesterov’s method).Convergent rate of FISTA becomes O(1/k2).

FISTA for ℓ1-regularization problemSet initial value x(0) and parameter γ > 0, then repeat

x(k+1) = Tγλ(z(k) − γDT(Dz(k) − y)

)t(k+1) =

1 +√

1 + 4t(k)2

2

z(k+1) = x(k+1) +t(k) − 1

t(k+1)(x(k+1) − x(k)) k = 0, 1, . . .


References I

M. Elad (2010)Sparse and Redundant Representations – From Theory to Applications inSignal and Image Processing –Springer, New York.

C. Bishop (2006)Pattern Recognition and Machine LearningSpringer-Verlag, New York.

S. Boyd and L. Vandenberghe (2004)Convex OptimizationCambridge University Press, Cambridge.

N. Parikh and S. Boyd (2013)Proximal AlgorithmsFoundations and Trends in Optimization, vol. 1, no. 3, pp. 123–231.


References II

P. Combettes and J.-C. Pesquet (2011)Proximal splitting methods in signal processingFixed-Point Algorithms for Inverse Problems in Science and Engineering,pp. 185–212.

山田功 (2009)工学のための関数解析数理工学社, Tokyo.冨岡亮太 (2015)スパース性に基づく機械学習講談社, Tokyo.永原正章 (2018)スパースモデリング –基礎から動的システムの応用 –コロナ社, Tokyo.


References III

関原謙介 (2015)ベイズ信号処理 –信号・ノイズ・推定をベイズ的に考える共立出版, Tokyo.


音響システム特論 第2回 advanced topics of acoustic systems day 2 · 2019-09-30 ·...

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音響システム特論第2回 advanced topics of acoustic systems day 2 · 2019-09-30 ·...