Organizing Gaussian mixture models into a tree for scaling up speaker retrieval

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Organizing Gaussian mixture models into a tree for

scaling up speaker retrieval

Jamal Rougui, Marc Gelgon, D. Aboutajdine, Noureddine Mouaddib, M.

Rziza

To cite this version:

Jamal Rougui, Marc Gelgon, D. Aboutajdine, Noureddine Mouaddib, M. Rziza. Organizing Gaussian mixture models into a tree for scaling up speaker retrieval. Pattern Recognition Letters, Elsevier, 2007, 28 (11), pp.1314-1319. �hal-00416675�

a,b a b a b

a

S1, . . . , SM

< O(M)

M M k Sk(x) = mk ! i=1 wi_kN_ki(x) Ni k(x) µik Σik wi k D M M S1 S2 S12

p(D|S12) p(D|S1) p(D|S2) S12 p(D|S12) p(D|S1) p(D|S2) m12 S12 m1+ m2 S1 S2 S₁₂ ESk[ ln p(D|Sk) ] − ESk[ ln p(D|S12) ] , k = 1, 2 " S12 " S12 = arg min S # − $ S1(x) ln S12(x) dx − $ S2(x) ln S12(x) dx % S KL(S1+2∥S12) S1+2(x) 1₂(S1(x) + S2(x)) " S12 = arg min_S & − $ S1+2(x) ln S12(x) S1+2(x) dx ' " S12 = arg min_S & − m1!+m2 i w₁₊₂i $ N₁₊₂i (x) ln S12(x) dx ' S12 Ni 1+2

KLm KL S1+2 S12 " S12 = arg min_S [KLm(S1+2∥S12)] = arg min S &_m1+m2 ! i=1 wi₁₊₂ minm12 j=1 KL(N i 1+2∥N j 12) ' KLm(Sk∥S12) = mk ! i=1 w_ki minm12 j=1 KL(N i k∥N j 12), k = 1, 2 (µ1, Σ1) (µ2, Σ2) 1 2(log |Σ2| |Σ1|+ T r(Σ −1 2 Σ1) + (µ1− µ2)TΣ−21(µ1− µ2) − δ) δ π _{m1 + m2 S1} m12 (< m1+ m2) S12 S1+2 S12 KLm S m1+ m2 m12 S₁₊₂ S₁₂ π0

π0 M× M S₁ S₂ KLm(S1||S2) + KLm(S2||S1) S1+2 S12 log₂(M) KLm

S12 ˆ π0 it= 0 S₁₂ ˆ πit " S12 it = arg min S12∈S m12 KLm(S1+2, S12,ˆπit) Sm12 m12 Mc S12 j ˆ w₁₂j = ! i∈π−1(j) w₁₊₂i ˆ µj₁₂ = ( i∈π−1(j)w i 1+2µi1+2 ˆ wj₁₂ ˆ Σj12 = ( i∈π−1(j)w i 1+2(Σi1+2+ (µi1+2− ˆµ j 12)(µi1+2− ˆµjr)T) ˆ w₁₂j π−₁ (j) ˆπ−_1,it (j) S₁₊₂ j S₁₂ ) S₁₂it πit+1 {1, . . . , m1+ m2} {1, . . . , m12} S1+2 S)12it ˆ

πit+1 _{= arg min}

π KLm(S1+2, " S12, π) i S1+2 j S)it 12

πit+1_{(i) = arg min}

j KL(N i 1+2||N j 12) πit+1_{= π}it

Bin. tree to N−Tree transformation

Binary tree of the GMMs speaker using similarity criterion

Grouping a GMMs Speaker according to bin. tree map Tree level ! Sp ∈ parents ! Sc ∈ Sp KLm(Sp∥Sc) KLm(Sparent∥Schild)

Sp {S1, S2, . . .} Sp log p(D|Sp) ≈ log p(D|Sc) log p(D|Sp) k log p(D|Sk) log p(D|Sp) log p(D|S˜ k) ≈ log p(D|Sp) + KL(Sp||Sk) , k= 1, 2, . . . KL(Sp||Sk) KL(Sp||Sk) KLm log p(D|S˜ k),

KLm

KLm KL_m

M inERR = min

k KL(Sp||Sk),

[log p(D|Sp) + MinERR

log p(D|Sp) + MaxERR]

• • KLm KLm KLm KLm KLm KL_m