Upload
ledat
View
226
Download
0
Embed Size (px)
Citation preview
Learning with matrix and tensor based models
using low-rank penalties
Johan Suykens
KU Leuven, ESAT-SCD/SISTAKasteelpark Arenberg 10
B-3001 Leuven (Heverlee), BelgiumEmail: [email protected]
http://www.esat.kuleuven.be/scd/
Nonsmooth optimization in machine learning, Liege, March 4 2013
(joint work with Marco Signoretto, Quoc Tran Dinh, Lieven De Lathauwer)
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens
Learning with matrices and tensorsneuroscience: EEG data
(time samples × frequency × electrodes)
computer vision: image (/video) compression/completion/· · ·(pixel × illumination × expression × · · ·)
web mining: analyze users behaviors
(users × queries × webpages)
vector x matrix X tensor X
data vector x −→ data matrix X −→ data tensor Xvector model: −→ matrix model: −→ tensor model:y = wTx y = 〈W,X〉 y = 〈W,X〉
[Signoretto M., Tran Dinh Q., De Lathauwer L., Suykens J.A.K., “Learning with Tensors:
a Framework Based on Convex Optimization and Spectral Regularization”, 2011]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 1
Overview
• Sparsity
• Matrix completion and tensor completion
• Learning with matrices and low rank penalty
• Learning with tensors
• Optimization algorithms
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 1
Learning with matrices and tensors
vector x matrix X tensor X
data vector x −→ data matrix X −→ data tensor Xvector model: −→ matrix model: −→ tensor model:y = wTx y = 〈W,X〉 y = 〈W,X〉
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 2
Sparsity in machine learning
• through loss function: model y =∑
i αiK(x, xi) + b
min wTw + γ∑
i
L(ei)
⇒ sparse α
• through regularization: model y = wTx + b
min∑
j
|wj| + γ∑
i
e2i
⇒ sparse w
−ε 0 +ε
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 3
Sparsity (1)
• Underdetermined linear system:
Ax = b A ∈ Rn×m, n < m
• Minimum norm solution:
minx
‖x‖22 s.t. Ax = b ⇒ x = AT (AAT )−1b
• Sparsest solution:
(P0) minx
‖x‖0 s.t. Ax = b (with ‖x‖0 = #i : xi 6= 0)
• Alternatives: lp-norms ‖x‖p = (∑
i |xi|p)1/p
(Pp) minx
‖x‖p s.t. Ax = b
Nonconvex for 0 < p < 1, convex for p = 1.
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 4
Sparsity (2)
• Mutual coherence: µ(A) = max1≤k,j≤m, k 6=j|aT
k aj|
‖ak‖2‖aj‖2
For a full rank A ∈ Rn×m, n < m, if a solution x exists satisfying
‖x‖0 <1
2(1 +
1
µ(A))
it is both the unique solution of (P1) and (P0).
• Restricted Isometry Property (RIP): Matrix A ∈ Rn×m has by
definition RIP(δ, k) if each submatrix AI (by combining at most kcolumns of A) has its nonzero singular values bounded between 1 − δand 1 + δ.
Matrix A with RIP(0.41; 2k) implies that (P1) and (P0) have identicalsolutions on all k-sparse vectors.
[Bruckstein et al., SIAM Review, 2009; Candes & Tao, 2005; Donoho & Elad, 2003; ...]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 5
Learning with matrices and tensors
vector x matrix X tensor X
data vector x −→ data matrix X −→ data tensor Xvector model: −→ matrix model: −→ tensor model:y = wTx y = 〈W,X〉 y = 〈W,X〉
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 6
Matrix completion: example
Given image (80 % missing entries)
[experiments by M. Signoretto]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 7
Matrix completion: example
Given image (80 % missing entries) and completed image
[experiments by M. Signoretto]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 7
Matrix completion: example
Given image (40 % missing entries)
[experiments by M. Signoretto]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 7
Matrix completion: example
Given image (40 % missing entries) and completed image
[experiments by M. Signoretto]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 7
Matrix completion: example
Original image
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 7
Matrix completion (1)
Given: matrix X with missing entriesGoal: complete the missing entriesAssumption: assume that X has low rank
minX
‖X‖∗
subject to Xij = Yij, i, j ∈ S
• given values Yij with i, j ∈ S a subset of all entries of the matrix
• Nuclear norm ‖X‖∗ =∑
i σi with σi the singular values of X(singular value decomposition: X =
∑
i σiuivTi )
• ‖X‖∗ is convex envelope of rankX on X : ‖X‖ ≤ 1 [Fazel, 2002]
• ‖X‖ ≤ ‖X‖F ≤ ‖X‖∗ ≤√
r‖X‖F ≤ r‖X‖ [Recht et al., 2010]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 8
Matrix completion (2)
This can be written as an SDP problem (semidefinite program):
minX,W1,W2
tr(W1) + tr(W2)
subject to Xij = Yij i, j ∈ S[
W1 XX∗ W2
]
0
The nuclear norm plays a similar role as the l1 norm, at the matrix level.
[Fazel et al., 2001; Candes & Recht, 2009]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 9
Matrix completion: RIP property
• Consider:(P0) : min rank X s.t. A(X) = b
(P1) : min ‖X‖∗ s.t. A(X) = b
• r-restricted isometry constant: smallest number δr(A) such that
1 − δr(A) ≤ ‖A(X)‖‖X‖F
≤ 1 + δr(A)
holds for all X of rank at most r, with A : Rm×n → R
p a linear map.
• Suppose that δ2r < 1 for integer r ≥ 1. Then solution to (P0) is theonly matrix of rank at most r satisfying A(X) = b.
• Suppose that r ≥ 1, is such that δ5r < 110, then solution to (P1) equals
solution to (P0).
[Recht, Fazel, Parrilo, Siam Rev, 2010]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 10
Tensor completion
Given: N -th order tensor X ∈ RI1×..×IN with missing entries
Goal: complete the missing entriesAssumption: assume that X has low rank
minX
‖X‖∗
subject to Xi1i2...iN = Yi1i2...iN , i1, i2, ..., iN ∈ Swith
• given entries Yi1i2...iN with i1, i2, ..., iN ∈ S a subset of the tensor
• Nuclear norm ‖X‖∗ = 1N
∑
n∈NN‖X〈n〉‖∗ with X〈n〉 the n-th mode
matrix unfolding
[Signoretto M., Van De Plas R., De Moor B., Suykens J.A.K., IEEE-SPL, 2011; Gandy et
al., 2011, Tomioka et al., 2011]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 11
Mass spectral imaging - digital staining
Data tensor: 51 × 34 pixels × 6490 variables/spectrumGiven partial labelling (4 classes), SVM prediction on all pixels
cerebellar cortex - Ammon’s horn section of hippocampus - cauda-putamen - lateral ventricle area
[Luts J., Ojeda F., Van de Plas R., De Moor B., Van Huffel S., Suykens J.A.K., ACA 2010]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 12
Tensor completion on mass spectral imaging
Mass spectral imaging: sagittal section mouse brain [data: E. Waelkens, R. Van de Plas]
Tensor completion using nuclear norm regularization [Signoretto et al., IEEE-SPL, 2011]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 13
Multichannel EEG for patient-specific seizure detection
• The electroencephalogram (EEG) measures the electrical activity ofthe brain and is a well-established technique in epilepsy diagnosis andmonitoring.
• Automatic seizure detection would drastically decrease the workloadof clinicians; EEG can provide accurate information about the onset ofthe seizure.
• As the seizure spreads quickly through the brain, the early detection ofthe seizure is essential.
[Hunyadi B., Signoretto M., Van Paesschen W., Suykens J., Van Huffel S., De Vos A.,
Clinical Neurophysiology, 2012]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 14
Feature-channel matrix
Extracted features:
Time domain features:1.-3. Number zero crossings, max & min
4. Skewness (skew)
5. Kurtosis (kurt)
6. Root mean square amplitude (rmsa)
Frequency domain features:7. Total power (TP)
8. Peak frequency (PF)
9.-16. Mean and normalized power in frequency bands:
delta: 13 Hz (D, nD), theta: 48 Hz (T, nT),
alpha: 913 Hz (A, nA), beta: 1420 Hz (B, nB)
EEG data: CHB-MIT database - scalp EEG recordings, 23 pediatric patients, 18 channels
0 2 4 6 8 10
T8−P8FT10−T8
FT9−FT10T7−FT9
P7−T7CZ−PZFZ−CZP8−O2T8−P8F8−T8
FP2−F8P4−O2C4−P4F4−C4
FP2−F4P3−O1C3−P3F3−C3
FP1−F3P7−O1T7−P7F7−T7
FP1−F7
Time (sec)
365 uV
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 15
Model with nuclear norm regularization
• Synchronization between EEG channels is a generally occurringcharacteristic. Representing the data in matrix form allows to exploit thecommon information among the channels.
• Model: (per patient)y = 〈W,X〉 + b
where 〈W,X〉 =∑
ij WijXij with X,W ∈ Rd×p,
d the number of features, p number of channels.Classifier with decision rule sign[y]
• Training from given data (Xk, yk)Nk=1:
minW,b
N∑
k=1
(yk − yk)2 + µ‖W‖∗
with nuclear norm ‖W‖∗ =∑
i σi with singular values σi; the labels±1 correspond to seizure and non-seizure epoch.
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 16
Multichannel EEG for patient-specific seizure detection
[Hunyadi B., Signoretto M., Van Paesschen W., Suykens J., Van Huffel S., De Vos A., Clinical Neurophysiology, 2012]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 17
Learning with matrices and tensors
vector x matrix X tensor X
data vector x −→ data matrix X −→ data tensor Xvector model: −→ matrix model: −→ tensor model:y = wTx y = 〈W,X〉 y = 〈W,X〉
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 18
Tensors
• N -th order tensor A ∈ RI1×I2×···×IN
• inner product: 〈A,B〉 :=∑
i1
∑
i2· · ·
∑
iNAi1i2···iNBi1i2···iN
• norm: ‖A‖ :=√
〈A,A〉• n−mode vector: obtained by varying in and keeping other indices fixed
• n−rank rankn(A): dimension of space spanned by n−mode vectors
• rank-(r1, r2, . . . , rN) tensor: tensor for which rn = rankn(A) for n ∈ NN
• multilinear rank: N−tuple (r1, r2, . . . , rN)
• rank: rank(A) := arg min
R ∈ N : A =∑
r∈NRu
(1)r ⊗ u
(2)r ⊗ · · · ⊗
u(N)r : u
(n)r ∈ R
In ∀ r ∈ NR, n ∈ NN
• property: rankn(A) ≤ rank(A) ∀n
• special case of matrix: rank1(A) = rank2(A) = rank(A)
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 19
Tensors
• N -th order tensor A ∈ RI1×I2×···×IN
• inner product: 〈A,B〉 :=∑
i1
∑
i2· · ·
∑
iNAi1i2···iNBi1i2···iN
• norm: ‖A‖ :=√
〈A,A〉• n−mode vector: obtained by varying in and keeping other indices fixed
• n−rank rankn(A): dimension of space spanned by n−mode vectors
• rank-(r1, r2, . . . , rN) tensor: tensor for which rn = rankn(A) for n ∈ NN
• multilinear rank: N−tuple (r1, r2, . . . , rN)
• rank: rank(A) := arg min
R ∈ N : A =∑
r∈NRu
(1)r ⊗ u
(2)r ⊗ · · · ⊗
u(N)r : u
(n)r ∈ R
In ∀ r ∈ NR, n ∈ NN
• property: rankn(A) ≤ rank(A) ∀n
• special case of matrix: rank1(A) = rank2(A) = rank(A)
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 20
Tensors
• N -th order tensor A ∈ RI1×I2×···×IN
• inner product: 〈A,B〉 :=∑
i1
∑
i2· · ·
∑
iNAi1i2···iNBi1i2···iN
• norm: ‖A‖ :=√
〈A,A〉• n−mode vector: obtained by varying in and keeping other indices fixed
• n−rank rankn(A): dimension of space spanned by n−mode vectors
• rank-(r1, r2, . . . , rN) tensor: tensor for which rn = rankn(A) for n ∈ NN
• multilinear rank: N−tuple (r1, r2, . . . , rN)
• rank: rank(A) := arg min
R ∈ N : A =∑
r∈NRu
(1)r ⊗ u
(2)r ⊗ · · · ⊗
u(N)r : u
(n)r ∈ R
In ∀ r ∈ NR, n ∈ NN
• property: rankn(A) ≤ rank(A) ∀n
• special case of matrix: rank1(A) = rank2(A) = rank(A)
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 21
Mode unfoldings of a tensor
• n−mode unfolding A〈n〉 ∈ RIn×J (matricization):
matrix whose columns are the n−mode vectors with J :=∏
j∈NN\n Ij
• n−mode unfolding ·〈n〉 : RI1×I2×···×IN → R
In×J .
• refolding: ·〈n〉 : RIn×J → R
I1×I2×···×IN
• property: rankn(A) = rank(A〈n〉)
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 22
Multilinear SVD (1)
[De Lathauwer L., De Moor B., Vandewalle J., 2000]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 23
Multilinear SVD (2)
• n−mode product A×n U ∈ RI1×I2×···×In−1×Jn×In+1×···×IN :
product of tensor A ∈ RI1×I2×···×IN by matrix U ∈ R
Jn×In
• multilinear SVD:
A = S ×1 U(1) ×2 U
(2) ×3 · · · ×N U(N)
with
– core tensor S ∈ RI1×I2×···×IN
– U(n) ∈ R
In×In a matrix of n−mode singular vectors, i.e., left singularvectors of the n−mode unfolding W〈n〉 with SVD
A〈n〉 = U(n)diag(σ(A〈n〉))V
(n)⊤
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 24
Inductive and transductive learning
transductive learning with tensors inductive learning with tensors
soft-completion
data: partially specified input data tensor
and matrix of target labels
data: pairs of fully specified input features
and vectors of target labels
output: latent features and missing labels output: models for out-of-sample evaluations
of multiple tasks
hard-completion
data: pairs of fully specified input features
and vectors of target labels
output: missing input data
[Signoretto M., Tran Dinh Q., De Lathauwer L., Suykens J.A.K., 2011]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 25
Inductive learning with tensors: setting
• Training data DN =(
X (n), y(n))
∈ RD1×D2×···×DM × R
T : n ∈ NN
n = 1, ..., N training datat = 1, ..., T outputs (tasks)M−th order input data tensor
• Modelyt = 〈W(t),X〉 + bt, t = 1, ..., T
• Assumptions:
– X = X + E with X a rank-(r1, r2, . . . , rM) tensor– for core tensors:
〈W(t),X〉 = 〈SW(t),SX 〉low multilinear rank in W(t) = SW(t) ×1 U1 ×2 U2 × · · · ×M UM
– target lables yt generated according to p(yt|yt) = 1/(1 + exp(−ytyt))
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 26
Inductive learning with tensors: training
• Penalized empirical risk minimization:
minW,b
fDN(W, b) +
∑
m∈NM+1
λm ‖W〈m〉‖∗
with misclassification error e.g. based on logistic loss:
fDN: (W, b) 7→
∑
n∈NN
∑
t∈NT
log(
1 + exp(
−y(n)t
(
〈X (n),W(t)〉 + bt
)))
• gives a predictive model, applicable to input data X beyond the trainingdata
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 27
Transductive learning with X and Y completion
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 28
Transductive learning with tensors: setting
• Tensors X ∈ RD1×D2×···×DM×N and Y = [y(1)y(2) · · · y(N)] ∈ R
T×N
• Missing entries both in X and Y:SX , SY: index sets of observed entries of X and Y
SSX, SSY
: sampling operators related to the index sets
• Implicit model:
y(n)t = 〈W(t),X (n)〉 + bt, t = 1, ..., T
• Assumptions:
– X = X + E with X a rank-(r1, r2, . . . , rM , rM+1) tensor– targets yt generated according to p(ytn|ytn) = 1/(1 + exp(−ytnytn))
– rank([
X〈M+1〉, Y⊤])
≤ rM+1 ≪ min(N, J + T )
with J =∏
j∈NMDj
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 29
Transductive learning with tensors: estimation
• Estimation of X , Y, b:
min(X ,Y ,b)∈V
fλ0(X , Y, b) +∑
m∈NM
λm‖X〈m〉‖∗ + λM+1
∥
∥
∥
[
X〈M+1〉, Y⊤]∥
∥
∥
∗
• objective function:
– V has module spaces(
RD1×D2×···×DM×N
)
×(
RT×N
)
×RT and inner
product 〈(X1, Y1, b1), (X2, Y2, b2)〉V = 〈X1, X2〉+ 〈Y1, Y2〉+ 〈b1, b2〉– objective
fλ0(X , Y, b) = fx(X ) + λ0fy(Y, b)
with fx : X 7→ ∑
p∈NPlx((ΩS
XX )p, z
xp)
fy : (Y, b) 7→ ∑
q∈NQly((ΩS
Y(Y + b ⊗ 1J))q, z
yq )
– losses e.g. lx : (u, v) 7→ 12(u − v)2, ly : (u, v) 7→ log(1 + exp(−uv))
– zx, zy are vectors of the observed entries
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 30
Transductive soft completion: Olivetti faces
original
true label: 5 predicted: 3 predicted: 5
input data matrix-sc tensor-sc
original
true label: 5 predicted: 3 predicted: 5
input data matrix-sc tensor-sc
original
true label: 3
input data matrix-sc tensor-sc
predicted: 3predicted: 3
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 31
Impainting color images by hard completion
original given image completed
Tensor: mode 1 and 2: pixel space, mode 3: 8-bit RGB color information
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 32
Impainting color images by hard completion
original given image completed
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 33
Impainting color images by hard completion
original given image completed
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 33
Impainting color images by hard completion
original given image completed
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 33
Impainting color images by hard completion
original given image completed
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 33
Optimization algorithm (1)
The learning problems are instances of the following convex optimizationproblem on an abstract vector space:
minw∈W
f(w) + g(w)
subject to w ∈ C
with- f : convex and differentiable functional- ∇f is Lf -Lipschitz:
‖∇f(w) −∇f(v)‖W ≤ Lf‖w − v‖W ∀ w, v ∈ W ;
- g: convex but possibly non-differentiable functional- C ⊆ W is a is non-empty, closed and convex set
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 34
Optimization algorithm (2)
• Problem restatement
minw∈W
h(w) = f(w) + g(w) + δC (w) , δC : w 7→
0, if w ∈ C
∞, otherwise
• Proximity operator
x(t+1) = proxτh
(
x(t))
with
proxτh : x 7→ arg minw∈W
h(w) +1
2τ‖w − x‖2 .
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 35
Optimization algorithm (3)
• Operator splitting approach:split h(w) = f(w) + g(w) + δC (w)into f(w) + δC (w) and non-smooth term g(w)
• Douglas-Rachford splitting:
y(k) = arg minx∈C
f (x) + 12τ
∥
∥x − w(k)∥
∥
2
W→ (solved inexactly)
r(k) = proxτg(2y(k) − w(k))
w(k+1) = w(k) + γ(k)(
r(k) − y(k))
• Projection onto C
Proof of convergence for sequence y(k)k
Stopping criterion based on h
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 36
Note: matrix case - Singular Value Thresholding
Given matrix Y , finding the solution to
minX
1
2‖X − Y ‖2
F + λ‖X‖∗
with λ > 0, is given by a shrinkage operation on singular values of Y :
proxtrλ (Y ) = U max(S − λI, 0)V T
[Cai et al., 2008; Tomioka et al., 2011]
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 37
Tensor case
• Learning problems involve the tensor modes:
∑
m∈NM+1
λm ‖W〈m〉‖∗
• Consider space W with cartesian product W1 × W2 × · · · × WI andinner product 〈x, y〉 =
∑
i∈NI〈xi, yi〉i.
• Assume function g : W → R defined by
g : (x1, x2, . . . , xI) 7→∑
i∈NI
gi(xi)
where for any i ∈ NI, gi : Wi → R is convex. Then we have:
proxg(x) =(
proxg1(x1),proxg2
(x2), · · · , proxgI(xI)
)
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 38
Duplication - transductive learning case
• Duplication of the tensors leads to considering the set:
C :=
(X[1], X[2], . . . , X[M ], X[M+1], Y , b) ∈ W : X[1] = X[2] = . . . = X[M+1]
• This gives the problem statement:
min(X[1],X[2],...,X[M ],X[M+1],Y ,b)∈W
f(X[1], . . . , X[M+1], Y , b) + g(X[1], . . . , X[M+1], Y )
subject to (X[1], . . . , X[M+1], Y , b) ∈ C
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 39
Prox and tensor modes
We apply
proxτg(X[1], . . . , X[M+1], Y ) =(
proxτλ1‖σ(·〈1〉)‖1(X[1]), · · · , proxτλM‖σ(·〈M〉)‖1
(X[M ]),Z1, Z2
)
where [Z1(X , Y ),Z2(X , Y )] is a partitioning of
Z(X , Y ) = Udiag(
proxτλM+1‖σ(·)‖1
([
X〈M+1〉, Y⊤]))
V⊤
with
proxλ‖σ(·〈n〉)‖1(W) =
(
U(n)diag(dλ)V (n)⊤
)〈n〉
and (dλ)i := max(σi(W〈n〉) − λ, 0).
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 40
Conclusions
• Sparsity:from vectors to matricesfrom matrices to tensors
• Transductive and inductive learning with matrices/tensors:going beyond matrix/tensor completion
• Further details:
Signoretto M., Tran Dinh Q., De Lathauwer L., Suykens J.A.K., “Learning with Tensors:
a Framework Based on Convex Optimization and Spectral Regularization”, 2011
• Software:https://sites.google.com/site/marcosignoretto/codeshttp://www.esat.kuleuven.be/sista/ADB/software.php
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 41
Acknowledgements
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 42
Thank you
Learning with matrix and tensor based models using low-rank penalties - Johan Suykens 43