Introduction : statistical models of matrix recovery
DenoisingTime series completion

Matrix factorization for time series analysis
Pierre Alquier

StatML – Imperial and Oxford
January 20, 2022

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Co-authors
Alquier, P. and Marie, N. (2019). Matrix Factorization for Multivariate Time Series Analysis. The
Electronic Journal of Statistics.

Alquier, P., Marie, N. and Rosier, A. (2021). Tight Risk Bound for High Dimensional Time Series
Completion. Preprint arXiv :2102.08178.

Nicolas Marie

Amélie Rosier

Université Paris Nanterre

ESME Sudria and Université Paris Nanterre

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Overview
many statistical problems or ML problems involve theestimation of a high-dimensional, low-rank matrix.
under an independence assumption on the observations,fast algorithms are known, with strong statisticalguarantees.
little theory in the case of time series / dependent
observations.
take-home message : the known algorithms can also be
used for time series safely.

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Contents
1

Introduction : statistical models of matrix recovery
Motivating examples
Reminder on SVD

2

Denoising
Model and estimation
Theory in the independent case
Theory for time series

3Time series completion
Model and estimation
Theory in the independent case
Theory for time series
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Motivating examples
Reminder on SVD

Contents
1

Introduction : statistical models of matrix recovery
Motivating examples
Reminder on SVD

2

Denoising
Model and estimation
Theory in the independent case
Theory for time series

3Time series completion
Model and estimation
Theory in the independent case
Theory for time series
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Motivating examples
Reminder on SVD

Some examples
Some stats/ML problems where the parameter is a large
matrix :1
denoising,2
reduced rank regression (RRR),
3
matrix completion (e.g for recommender systems),
4
quantum tomography,
5community detection in graphs...
Inzenman, A. J. (1975). Reduced-rank regression for the multivariate linear model. Journal ofMultivariate Analysis.
Candès, E. and Tao, T. (2010). The power of convex relaxation : Near-optimal matrix
completion. IEEE Transactions on Information Theory.
Gross, D., Liu, Y. K., Flammia, S. T., Becker, S. and Eisert, J. (2010). Quantum state
tomography via compressed sensing. Physical review letters.
Paul, S. and Chen, Y. (2020). Spectral and matrix factorization methods for consistentcommunity detection in multi-layer networks. The Annals of Statistics.
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Motivating examples
Reminder on SVD

Dimension reduction
in denoising, M is d × T and we observe exactlyn = d × T entries (with noise),in matrix completion, M is d × T and we observe only nentries with n  d × T (possibly with noise)...
Estimation is only possible under dimension reduction.
The low-rank assumption is meaningful in many examples (andsuccessful in applications).

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Motivating examples
Reminder on SVD

Singular Value Decomposition (SVD)
Assume that M is d × T and rank(M) = r .





V1T

 σ1 0 0




..
  0 ... 0 
U
.
.
.
U
M =


1
r
.




T
0 0 σr
Vr
{z
}|
{z
}
|
=Σ (r ×r )|{z}
=V T (r ×T )=U (d×r )M = (UΣ) |{z}V T = |{z}
W |{z}VT .
| {z }
d×r

r ×T

d×r

r ×T

We have ' r (d + T ) parameters to estimate.
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Motivating examples
Reminder on SVD

Existing results
Efficient algorithms exploiting a low-rank assumption were
proposed.
A strong theory was developped so these algorithms come withguarantees. For example :in the noiseless caseCandès, E. and Tao, T. (2010). The power of convex relaxation : Near-optimal matrix
completion. IEEE Transactions on Information Theory.

when the noise is i.i.d
Koltchinskii, V., Lounici, K. and Tsybakov, A. (2011). Nuclear-norm penalization and optimal
rates for noisy low-rank matrix completion. The Annals of Statistics.

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Contents
1

Introduction : statistical models of matrix recovery
Motivating examples
Reminder on SVD

2

Denoising
Model and estimation
Theory in the independent case
Theory for time series

3Time series completion
Model and estimation
Theory in the independent case
Theory for time series
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Matrix denoising model

Matrix denoising model
We observe X where
X =M +E
and
M is a d × T matrix (to recover),E is some random noise.

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Estimation
Reminder : Frobenius norm and nuclear normsX
X
kAkF =
A2i,j , kAk1 =σi (A).
i,ji

Rank constrained estimator :bk = arg min kX − Ak2 .
MFrank(A)≤k
Solution : in the SVD of X , replace each σi (X ) by σi (X )1{i>k} .

Nuclear-norm penalized estimator :eλ = arg min kX − Ak2 + λkAk1 .
M
FASolution : in the SVD of X , replace each σi (X ) by max(σi (X ) − λ, 0).
2
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Theoretical analysis
We first focus on the independent case : the entries of E are
i.i.d N (0, σ 2 ).
Theorem
For any s > 0, with probability at least 1 − 2 exp(−s),


2
2
2
b
kMk − MkF ≤ C
inf kA − MkF + σ k(d + T + s)rank(A)≤kfor some universal constant C . In particular, if
rank(M) = r ≤ k,bk − Mk2 ≤ C σ 2 k(d + T + s).
kM
F
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

A toy example : denoising Mondrian’s paintings

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Mondrian’s paintings : black and white version

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Mondrian’s paintings : adding a Gaussian noise

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Mondrian’s paintings : SVD of X
> SVD = svd(X)> SVD$d[1] 705.2966509 147.2653999 40.3910584
6.1156832[5]5.9563710
5.8878358
5.8747333
5.8484616[10]5.8154512
5.7746063
5.7607876
5.7512011[15]
5.7245453
5.7007277
5.6785695
5.6556892...
[765]
0.4789888
0.4684564
0.4519168
0.4447403> SVD$d[4:768] = 0> Mhat = SVD$u %*% diag(SVD$d) %*% t(SVD$v)

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Mondrian’s paintings : adding a Gaussian noise

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Rank selection
Theorem
For some universal constant c > 0 (see the paper) we definenobk − X k2 + cσ 2 k(d + T + s) .
rb = arg min kM
FkThen, if rank(M) = r unknown, with probability at least1 − 2 exp(−s),
brb − Mk2 ≤ C σ 2 r (d + T + s).
kM
F

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time seriesMultivariate time series matrices
x1,1 x1,2 . . .
X =  x2,1 x2,1 . . . 
x3,1 x3,1 . . .
Shih, S.-Y., Sun, F.-K. and Lee, H.-Y. (2019). Temporal pattern attention for multivariate timeseries forecasting. Machine Learning.
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Theory for time seriesThis time, we allow the rows of E to be N (0, Σ).
Theorem
For any s > 0, with probability at least 1 − 2 exp(−s),
22bk −Mk ≤ C
kM
inf kA − MkF + kΣkop k(d + T + s)Frank(A)≤k

where kΣkop = supx kΣxk/kxk is the operator norm of Σ.
Examples :i.i.d noise, Σ = σ 2 I , then kΣkop = σ 2 .
autoregressive noise Ei,t+1 = ρEi,t + ηi,t , Var(ηi,t ) = σ 2 , then



2
Σ=σ 



1
ρ
.
.
.
ρT −1

ρ
1

...
...

.

.

.

.

...

.

.

ρ

Pierre Alquier, RIKEN AIP

ρT −1
ρT −2
.
.
.
1





2 1 + |ρ|.
 and kΣkop = σ

1 − |ρ|

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Contents
1

Introduction : statistical models of matrix recovery
Motivating examples
Reminder on SVD

2

Denoising
Model and estimation
Theory in the independent case
Theory for time series

3Time series completion
Model and estimation
Theory in the independent case
Theory for time series
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

A matrix completion model

Matrix completion model
We observe, for ` = 1, . . . , n(i` , j` ) drawn uniformly on {1, . . . , d} × {1, . . . , T },X` = Mi` ,j` + E` ,where
M is a d × T matrix (to recover),E` is some noise.

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Estimators
Rank constrained estimation :bk = arg min
MnX(X` − Ai` ,j` )2 .

rank(A)≤k `=1

Nuclear-norm penalized estimator :
( n
)
X
2
eλ = arg min
M(X` − Ai ,j ) + λkAk1 .
A

` `

`=1

Both estimators are implemented in R in the SoftImpute
package.
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Collaborative filtering / recommender systems

ClaireNial
BrendonAndrew
AdrianPierre
..
.

4
?
2
?
1
?
..
.

Pierre Alquier, RIKEN AIP

?
4
?
4
?
5
..
.

3
?
4
?
?
?
..
.

...
...
...
...
...
...
...

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Netflix Prize

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

http ://movielens.org

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Movies in MovieLens 100K
1682 movies

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Users in MovieLens 100K
943 users

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Reading the data with SoftImpute> X = read.table("u.data",header=FALSE)> X
V1 V2 V3V4
1 196 242 3 881250949
2 186 302 3 891717742
3
22 377 1 878887116
4 244
51 2 880606923
5 166 346 1 886397596....
> A = Incomplete(i=X$V1,j=X$V2,x=X$V3)

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Incomplete matrices> A[1,] 5 3 4 3 3 5 4 1 5 3 2 5 5 5 5 5 3 4 5 4[2,] 4 . . . . . . . . 2 . . 4 4 . . . . 3 .
[3,] . . . . . . . . . . . . . . . . . . . .
[4,] . . . . . . . . . . 4 . . . . . . . . .
[5,] 4 3 . . . . . . . . . . . . . . 4 . . .
[6,] 4 . . . . . 2 4 4 . . 4 2 5 3 . . . 4 .
[7,] . . . 5 . . 5 5 5 4 3 5 . . . . . . . .
[8,] . . . . . . 3 . . . 3 . . . . . . . . .
[9,] . . . . . 5 4 . . . . . . . . . . . . .
[10,] 4 . . 4 . . 4 . 4 . 4 5 3 . . 4 . . . .
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

The softImpute functionb ← minM
A(
X)(Ai,j − Xi,j )2 + λkAk1i,j

How it works :> B = softImpute(A, rank.max = ..., lambda = ...,
type = x("als","svd"), ... )> i = ...
> j = ...
> impute(B,i,j)

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Example> A[1,] 5 3 4 3 3 5 4 1 5 3 2 5 5 5 5 5 3 4 5 4[2,] 4 . . . . . . . . 2 . . 4 4 . . . . 3 .
[3,] . . . . . . . . . . . . . . . . . . . .
[4,] . . . . . . . . . . 4 . . . . . . . . .
> B=softImpute(A,rank.max=5,lambda=0,type="svd")> impute(B,3,1)[1] 1.361311> impute(B,4,1)[1] 2.48563

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

How to get many entries> A[1,] 5 3 4 3 3 5 4 1 5 3 2 5 5 5 5 5 3 4 5 4[2,] 4 . . . . . . . . 2 . . 4 4 . . . . 3 .
[3,] . . . . . . . . . . . . . . . . . . . .
[4,] . . . . . . . . . . 4 . . . . . . . . .
> B=softImpute(A,rank.max=5,lambda=0,type="svd")> i = c(3,3,3,4,4,4)> j = c(1,2,3,1,2,3)> impute(B,i,j)[1] 1.3432932 0.3631449 1.0265744 2.7306180
0.1185635 1.8732696
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

How to get the full matrix> B=softImpute(A,rank.max=5,lambda=0,type="svd")> Y = complete(A,B)
Be careful, it can be larger than the memory of your laptop !

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Completed MovieLens matrix> round(Y)[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] ...
[1,]
5
3
4
3
3
5
4
1
5
[2,]
4
1
2
2
0
3
4
3
5
[3,]
1
0
1
2
1
1
2
1
3
[4,]
3
0
2
2
1
3
4
2
5
...
> A
[1,] 5 3 4 3 3 5 4 1 5 3 2 5 5 5 5 5 3 4 5 4[2,] 4 . . . . . . . . 2 . . 4 4 . . . . 3 .
[3,] . . . . . . . . . . . . . . . . . . . .
[4,] . . . . . . . . . . 4 . . . . . . . . .
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

MSE by rank
> data = X[1:80000,]> A = Incomplete(i=data$V1,j=data$V2,x=data$V3)> test = X[80001:100000,]> MSE = c()> for (k in 1:10)> {> B = softImpute(A,rank.max=k,lambda=0,type="svd")> pred = impute(object=B,i=test$V1,j=test$V2)> MSE = c(MSE,mean((pred-test$V3)^2))> }Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

MSE by rank

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Minimax rate of estimation
TheoremAssume the noise (ε` ) is i.i.d N (0, σ 2 ) (sub-exponential also
works). For a well chosen λ that does not depend onr = rank(M), and under minimal assumptions on M, withlarge probability221eλ − M ≤ C σ r (d + T ) log(d + T ) .
M
dT
n
F

Moreover, this rate is minimax-optimal (up to the log term).
Koltchinskii, V., Lounici, K. and Tsybakov, A. (2011). Nuclear-norm penalization and optimal
rates for noisy low-rank matrix completion. The Annals of Statistics.

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time seriesMultivariate time series

Imperio, S. et al. (2010). Investigating
population dynamics in ungulates : Do
hunting statistics make up a good index
of population abundance ? Wildlife
Biology.

multivariate seriescorrelations
noisy observations
missing entries

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Exampleseconometrics : panel data with missing entries,
industry : data from sensors at multiple locations,
ecology : spatial data with observations from a few sites
only at each date,...
more generally, any situation where we have multivariate
time series and each measurement is expensive.

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Assumptions
low-rank trend :M = |{z}U |{z}V
d×k k×T

temporal correlated
noise ε :εi,t indep. εj,t 0 (i 6= j)εi,t not indep. εi,t 0(i` , t` ) i.i.d uniform, ξ`observation noise :X` = Mi` ,t` + εi` ,t` + ξ` .
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time seriesReminder on φ-mixing
Given two σ-algebras A and B,noφ(A, B) = sup |P(A)−P(A|B)| : (A, B) ∈ A×B, P(B) > 0 .
Given a stationary time series S = (St )t∈Z ,φS (h) = φ σ(. . . , St−2 , St−1 , St ), σ(St+h , St+h+1 , St+h+2 , . . . ) .
The series S is said to be φ-mixing iff∞
X

φS (h) < +∞.

h=0
Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

More assumptions
X` = Mi` ,t` + εi` ,t` + ξ` .
√M = |{z}U |{z}V and |Ui,h |, |Vh,t | ≤ cU,V / r .
d×r

r ×T

(i` , t` ) i.i.d uniform on {1, . . . , d} × {1 . . . , T } ;(εi,t )t=1,...,T is a bounded, φ-mixing time series :|εi,t | ≤ mε and∞X

φεi,· (t) ≤ Φε .

t=1(ξ` ) are i.i.d, sub-exponential variables : for k ≥ 2,vξ cξq−2 q!
E(|ξ` | ) ≤.
2
q

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Estimator and risk bound
bk =
M

n
X

(X` − Ai` ,j` )2 .
arg min
A =|{z}W |{z}
Z `=1
|{z}
d×T

d×k

k×T

TheoremWith probability at least 1 − exp(−s), as soon as k ≥ r ,2
1
bk − M ≤ C k(d + T ) log(n) + s
M
dT
n
F

where C = C (cU,V , mε , Φε , vξ , cξ ) is known.

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Remarks on the proof
1

2

decompose the P
difference between empirical
P risk andn1
12
expected risk n `=1 (Y` − Xi` ,j` ) − dT i,j (Mi,j − Xi,j )2
in elementary terms.
some of these terms are sums of i.i.d variables. Boundthem via Bernstein inequality. Some are sums of φ-mixing
variables, use :Samson, P.-M. (2000). Concentration of measure inequalities for Markov chains and Φ-mixing
processes. The Annals of Probability.

3

union bound.

REMARK : if the εi,· satisfy another notion of mixing or weak-dependence, we can use alternative√
versions of Bernstein inequality but this lead to slower rates of convergence, in 1/ n.

Pierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

Rank selection(
rb =arg min1≤k≤min(d,T )n

1X bk(d + T ) log(n)((Mk )i` ,j` − X` )2 + c
n `=1
n

)

where c = c(cU,V , mε , Φε , vξ , cξ ) is known but too large.
In practice : we use the slope heuristic to calibrate a better c.
TheoremWith probability at least 1 − exp(−s),
21brˆ − M ≤ C r (d + T ) log(n) + s .
M
dTnFPierre Alquier, RIKEN AIP

Matrix factorization for time series

Introduction : statistical models of matrix recovery
DenoisingTime series completion

Model and estimation
Theory in the independent case
Theory for time series

What about Bayes ?
Review of low-rank inducing priors, and theoretical study fordenoising / matrix regression :Alquier, P. (2013). Bayesian Methods for Low-rank Matrix Estimation : Short Survey andTheoretical Study. International conference on Algorithmic Learning Theory (ALT).

For matrix completion :Mai, T. T. and Alquier, P. (2015). A Bayesian Approach for Matrix Completion : Optimal Rateunder General Sampling Distribution. Electronic Journal of Statistics.

Pierre Alquier, RIKEN AIP

Matrix factorization for time series