FILTER COEFFICIENT OPTIMIZATION APPARATUS, LATENT VARIABLE OPTIMIZATION APPARATUS, FILTER COEFFICIENT OPTIMIZATION METHOD, LATENT VARIABLE OPTIMIZATION METHOD, AND PROGRAM
Provided is a technology of optimizing a latent variable by solving a convex optimization problem equivalent to a non-convex optimization problem instead of solving the non-convex optimization problem. A latent variable optimization apparatus includes an optimization unit that calculates an optimum value ˜w* of a latent variable ˜w based on an optimization problem min ˜w (L convex (˜w)+Σ d=1 D L d (˜w)), L convex being a strongly convex function relevant to the latent variable ˜w, L d being a function relevant to the latent variable ˜w, S d,1 , . . . , S d,C being a region that is obtained by dividing a domain of the function L d into C closed convex sets, ∧ d,c being a convex function that is defined on the region S d,c and that approximates the function L d , c d being a discrete variable that has a value of 1, . . . , C, the optimization unit calculating the optimum value ˜w* by solving an optimization problem min c_1, . . . , c_D (min ˜w (L convex (˜w)+Σ d=1 D ∧ d,c_d (˜w))) instead of solving the above optimization problem.
1 . A filter coefficient optimization apparatus including an optimization unit that calculates an optimum value w* of a filter coefficient w={w 1 , . . . , w F } (w f (f=1, . . . , F, F is an integer equal to or more than 1) is a filter coefficient of a frequency bin f) of a beamformer that emphasizes sound (hereinafter referred to as target sound) from D sound sources (hereinafter referred to as a sound source 1, . . . , a sound source D),
D being an integer equal to or more than 1,
R f (f=1, . . . , F) being a spatial correlation matrix for sound other than the target sound relevant to the frequency bin f, L MV_f (w f )=w f H R f w f (f=1, . . . , F) being a cost function relevant to a filter coefficient w f ,
the optimization unit calculating the optimum value w* based on an optimization problem min w_1, . . . ,W_F Σ f=1 F L MV_f (w f ) relevant to the filter coefficient w under a predetermined constraint condition,
the predetermined constraint condition not including a constraint relevant to a phase of the filter coefficient w f (f=1, . . . , F).
2 . The filter coefficient optimization apparatus according to claim 1 , wherein:
θd (d=1, . . . , D) is an angular direction in which a sound source d exists, and a f,d (f=1, . . . , F, d=1, . . . , D) is an array manifold vector in the frequency bin f corresponding to a sound wave that comes from the angular direction Od, the sound wave being a plane wave; and
the predetermined constraint condition is expressed by the following expression:
[Math. 23]
| w f H a f,d |=1
(f=1, F, . . . ,d=1, D).
3 . The filter coefficient optimization apparatus according to claim 1 , wherein:
θd (d=1, . . . , D) is an angular direction in which a sound source d exists, and a f,d (f=1, . . . , F, d=1, D) is an array manifold vector in the frequency bin f corresponding to a sound wave that comes from the angular direction Od, the sound wave being a plane wave; and
the predetermined constraint condition is expressed by the following expression:
[Math. 24]
| w f H a f,d |≥1.
(f=1, F, d=1, D).
4 . The filter coefficient optimization apparatus according to claim 3 , wherein:
C is an integer equal to or more than 1, C f,d (f=1, . . . , F, d=1, . . . , D) is a discrete variable that has a value of 1, . . . , C, c f =(c f,i , . . . , c f,D ) (f=1, F) is a discrete variable that is defined by a discrete variable c f,i , . . . , c f,D , and ∧ (f,d),c_f,d (f=1, . . . , F, d=1, . . . , D) is a function relevant to a variable γ f,d that is defined by the following expression (γ f,d =w f H a f,d ):
[
Math
.
25
]
Λ
(
f
,
d
)
,
c
f
,
d
(
γ
f
,
d
)
=
{
0
(
R
(
γ
f
,
d
e
-
2
π
j
(
c
f
,
d
+
1
)
/
2
C
)
≥
1
,
2
π
c
f
,
d
C
≤
∠γ
f
,
d
≤
2
π
(
c
f
,
d
+
1
)
C
)
∞
(
otherwise
)
;
and
the optimization unit calculates the optimum value w*, by solving an optimization problem min {c_f,w_f} (Σ f=1 F L MV_f (w f )+Σ f=1 F Σ d=1 D ∧ (f,d),c_f,d (w f H a f,d )) relevant to the filter coefficient w and the discrete variable c 1 , . . . , C F instead of solving the optimization problem min w_1, . . . ,W_F Σ f=1 F L MV_f (w f ).
5 . The filter coefficient optimization apparatus according to claim 4 , wherein
the optimization unit includes a candidate calculation unit configured to calculate a candidate w f candidate [(c f,i , . . . , c f,D )] of the optimum value of the filter coefficient w f for all values that the discrete variable (c f,1 , . . . , c f,D ) can have, for each frequency bin f, by the following expression:
[
Math
.
26
]
w
f
candidate
[
(
c
f
,
1
,
…
,
c
f
,
D
)
]
←
arg
min
w
f
(
L
MV
f
(
w
f
)
+
∑
d
=
1
D
Λ
(
f
,
d
)
,
c
f
,
d
(
w
f
H
a
f
,
d
)
)
and
an optimum value determination unit configured to adopt a candidate that is of the candidate w f candidate [(c f,1 , . . . , c f,D )] and that minimizes a value of a cost function L MV_f (w f )+Σ d=1 D ∧ (f,d),c_f,d (w f H a f,d ), as an optimum value w f * of the filter coefficient w f , for the frequency bin f, and configured to obtain the optimum value w* from w*={Cw 1 *, . . . , w F *}.
6 . A latent variable optimization apparatus including an optimization unit that calculates an optimum value ˜w* of a latent variable ˜w based on an optimization problem min ˜w (L convex (˜w)+Σ d=1 D L d (˜w)) relevant to the latent variable ˜w,
L convex being a strongly convex function relevant to the latent variable ˜w, L d (d=1, . . . , D, D is an integer equal to or more than 1) being a function relevant to the latent variable ˜w,
C being an integer equal to or more than 1, S d,1 , . . . , S d,C (d=1, D) being a region that is obtained by dividing a domain of the function L d into C closed convex sets, ∧ d,c (d=1, . . . , D, c=1, . . . , C) being a convex function that is defined on the region S d,c and that approximates the function L d , c d (d=1, . . . , D) being a discrete variable that has a value of 1, C,
the optimization unit calculating the optimum value ˜w* by solving an optimization problem min c_1, . . . ,c_D (min ˜w (L convex (˜w)+Σ d=1 D ∧ d,c_d (˜w))) relevant to the latent variable ˜w and the discrete variable c 1 , . . . , c D instead of solving the optimization problem min ˜w (L convex (˜w)+Σ d=1 D L d (˜w)).
7 . The latent variable optimization apparatus according to claim 6 , wherein
the optimization unit includes
a candidate calculation unit configured to calculate a candidate ˜w candidate [(c 1 , . . . , c D )] of the optimum value of the latent variable ˜w for all values that the discrete variable (c 1 , . . . , c D ) can have, by the following expression:
[
Math
.
27
]
w
~
candidate
[
(
c
1
,
…
,
c
D
)
]
←
arg
min
w
~
(
L
convex
(
w
~
)
+
∑
d
=
1
D
Λ
d
,
c
d
(
w
~
)
)
and
an optimum value determination unit configured to adopt a candidate that is of the candidate ˜w candidate [(c 1 , . . . , c D )] and that minimizes a value of a cost function L convex (˜w)+Σ d=1 D ∧ d,c_d (˜w), as the optimum value ˜w*.
8 . A filter coefficient optimization method including an optimization step in which a filter coefficient optimization apparatus calculates an optimum value w* of a filter coefficient w={w 1 , . . . , w F } (w f (f=1, . . . , F, F is an integer equal to or more than 1) is a filter coefficient of a frequency bin f) of a beamformer that emphasizes sound (hereinafter referred to as target sound) from D sound sources (hereinafter referred to as a sound source 1, . . . , a sound source D),
D being an integer equal to or more than 1,
R f (f=1, . . . , F) being a spatial correlation matrix for sound other than the target sound relevant to the frequency bin f, L MV_f (w f )=w f H R f w f (f=1, . . . , F) being a cost function relevant to a filter coefficient w f ,
the optimization step being a step of calculating the optimum value w* based on an optimization problem min w_1, . . . , W_F Σ f=1 F L MV_f (w f ) relevant to the filter coefficient w under a predetermined constraint condition,
the predetermined constraint condition not including a constraint relevant to a phase of the filter coefficient w f (f=1, . . . , F).
9 . A latent variable optimization method including an optimization step in which a latent variable optimization apparatus calculates an optimum value ˜w* of a latent variable ˜w based on an optimization problem min ˜w (L convex (˜w)+Σ d=1 D L d (˜w)) relevant to the latent variable ˜w,
L convex being a strongly convex function relevant to the latent variable ˜w, L d (d=1, . . . , D, D is an integer equal to or more than 1) being a function relevant to the latent variable ˜w,
C being an integer equal to or more than 1, S d,1 , . . . , S d,C (d=1, . . . , D) being a region that is obtained by dividing a domain of the function L d into C closed convex sets, ∧ d,c (d=1, . . . , D, c=1, . . . , C) being a convex function that is defined on the region S d,c and that approximates the function L d , ca (d=1, D) being a discrete variable that has a value of 1, C,
the optimization step being a step of calculating the optimum value ˜w* by solving an optimization problem min c_1, . . . , c_D (min ˜w (L convex (˜w)+Σ d=1 D ∧ d,c_d (˜w))) relevant to the latent variable ˜w and the discrete variable c 1 , . . . , C D instead of solving the optimization problem min ˜w (L convex (˜w)+Σ d=1 D L d (˜w)).
10 . A non-transitory computer-readable recording medium storing a program that causes a computer to function as the filter coefficient optimization apparatus according to claim 1 .
11 . A non-transitory computer-readable recording medium storing a program that causes a computer to function as the latent variable optimization apparatus according to claim 6 .