Robust Matrix Completion

Introduction

In the recent years, there have been a considerable interest in statistical inference for high-dimensional matrices. One particular problem is matrix completion where one observes only a small number $N\ll m_{1}m_{2}$ of the entries of a high-dimensional $m_{1}\times m_{2}$ matrix $L_{0}$ of rank $r$ and aims at inferring the missing entries. In general, recovery of a matrix from a small number of observed entries is impossible, but, if the unknown matrix has low rank, then accurate and even exact recovery is possible. In the noiseless setting, established the following remarkable result: assuming that the matrix $L_{0}$ satisfies some low coherence condition, this matrix can be recovered exactly by a constrained nuclear norm minimization with high probability from only $N\gtrsim r\max\{m_{1},m_{2}\}\log^{2}(m_{1}+m_{2})$ entries observed uniformly at random. A more common situation in applications corresponds to the noisy setting in which the few available entries are corrupted by noise. Noisy matrix completion has been in the focus of several recent studies (see, e.g., ).

The matrix completion problem is motivated by a variety of applications. An important question in applications is whether or not matrix completion procedures are robust to corruptions. Suppose that we observe noisy entries of $A_{0}=L_{0}+S_{0}$ where $L_{0}$ is an unknown low-rank matrix and $S_{0}$ corresponds to some gross/malicious corruptions. We wish to recover $L_{0}$ but we observe only few entries of $A_{0}$ and, among those, a fraction happens to be corrupted by $S_{0}$. Of course, we do not know which entries are corrupted. It has been shown empirically that uncontrolled and potentially adversarial gross errors affecting only a small portion of observations can be particularly harmful. For example, Xu et al. showed that a very popular matrix completion procedure using nuclear norm minimization can fail dramatically even if $S_{0}$ contains only a single nonzero column. It is particularly relevant in applications to recommendation systems where malicious users try to manipulate the outcome of matrix completion algorithms by introducing spurious perturbations $S_{0}$. Hence, there is a need for new matrix completion techniques that are robust to the presence of corruptions $S_{0}$.

A particular case of this setting is the matrix decomposition problem where $N=m_{1}m_{2}$, i.e., we observe all entries of $A_{0}$. Several recent works consider the matrix decomposition problem, mostly in the noiseless setting, $\xi_{i}\equiv 0$. Chandrasekaran et al. analyzed the case when the matrix $S_{0}$ is sparse, with small number of non-zero entries. They proved that exact recovery of $(L_{0},S_{0})$ is possible with high probability under additional identifiability conditions. This model was further studied by Hsu et al. who give milder conditions for the exact recovery of $(L_{0},S_{0})$. Also in the noiseless setting, Candes et al. studied the same model but with positions of corruptions chosen uniformly at random. Xu et al. studied a model, in which the matrix $S_{0}$ is columnwise sparse with sufficiently small number of non-zero columns. Their method guarantees approximate recovery for the non-corrupted columns of the low-rank component $L_{0}$. Agarwal et al. consider a general model, in which the observations are noisy realizations of a linear transformation of $A_{0}$. Their setup includes the matrix decomposition problem and some other statistical models of interest but does not cover the matrix completion problem. Agarwal et al. state a general result on approximate recovery of the pair $(L_{0},S_{0})$ imposing a “spikiness condition” on the low-rank component $L_{0}$. Their analysis includes as particular cases both the entrywise corruptions and the columnwise corruptions.

The robust matrix completion setting, when $N<m_{1}m_{2}$, was first considered by Candes et al. in the noiseless case for entrywise sparse $S_{0}$. Candes et al. assumed that the support of $S_{0}$ is selected uniformly at random and that $N$ is equal to $0.1m_{1}m_{2}$ or to some other fixed fraction of $m_{1}m_{2}$. Chen et al. considered also the noiseless case but with columnwise sparse $S_{0}$. They proved that the same procedure as in can recover the non-corrupted columns of $L_{0}$ and identify the set of indices of the corrupted columns. This was done under the following assumptions: the locations of the non-corrupted columns are chosen uniformly at random; $L_{0}$ satisfies some sparse/low-rank incoherence condition; the total number of corrupted columns is small and a sufficient number of non-corrupted entries is observed. More recently, Chen et al. and Li considered noiseless robust matrix completion with entrywise sparse $S_{0}$. They proved exact recovery of the low-rank component under an incoherence condition on $L_{0}$ and some additional assumptions on the number of corrupted observations.

To the best of our knowledge, the present paper is the first study of robust matrix completion with noise. Our analysis is general and covers in particular the cases of columnwise sparse corruptions and entrywise sparse corruptions. It is important to note that we do not require strong assumptions on the unknown matrices, such as the incoherence condition, or additional restrictions on the number of corrupted observations as in the noiseless case. This is due to the fact that we do not aim at exact recovery of the unknown matrix. We emphasize that we do not need to know the rank of $L_{0}$ nor the sparsity level of $S_{0}$. We do not need to observe all entries of $A_{0}$ either. We only need to know an upper bound on the maximum of the absolute values of the entries of $L_{0}$ and $S_{0}$. Such information is often available in applications; for example, in recommendation systems, this bound is just the maximum rating. Another important point is that our method allows us to consider quite general and unknown sampling distribution. All the previous works on noiseless robust matrix completion assume the uniform sampling distribution. However, in practice the observed entries are not guaranteed to follow the uniform scheme and the sampling distribution is not exactly known.

We establish oracle inequalities for the cases of entrywise sparse and columnwise sparse $S_{0}$. For example, in the case of columnwise corruptions, we prove the following bound on the normalized Frobenius error of our estimator $(\hat{L},\hat{S})$ of $(L_{0},S_{0})$: with high probability

This paper is organized as follows. Section 2.1 contains the notation and definitions. We introduce our estimator in Section 2.2 and we state the assumptions on the sampling scheme in Section 2.3. Section 3 presents a general upper bound for the estimation error. In Sections 4 and 5, we specialize this bound to the settings with columnwise corruptions and entrywise corruptions, respectively. In Section 6, we prove that our estimator is minimax rate optimal up to a logarithmic factor. The Appendix contains the proofs.

Preliminaries

General notation. For any set $I$, $|I|$ denotes its cardinality and $\bar{I}$ its complement. We write $a\vee b=\max(a,b)$ and $a\wedge b=\min(a,b)$.

For a matrix $A$, $A^{i}$ is its $i$th column and $A_{ij}$ is its $(i,j)-$th entry. Let $I\subset\{1,\dots m_{1}\}\times\{1,\dots m_{2}\}$ be a subset of indices. Given a matrix $A$, we denote by $A_{I}$ its restriction on $I$, that is, $\left(A_{I}\right)_{ij}=A_{ij}$ if $(i,j)\in I$ and $\left(A_{I}\right)_{ij}=0$ if $(i,j)\not\in I$. In what follows, $\mathbf{Id}$ denotes the matrix of ones, i.e., $\mathbf{Id}_{ij}=1$ for any $(i,j)$ and ${\bf 0}$ denotes the zero matrix, i.e., ${\bf 0}_{ij}=0$ for any $(i,j)$.

For any $p\geq 1$, we denote by $\|\cdot\|_{p}$ the usual $l_{p}-$norm. Additionally, we use the following matrix norms: $\|A\|_{*}$ is the nuclear norm (the sum of singular values), $\|A\|$ is the operator norm (the largest singular value), $\|A\|_{\infty}$ is the largest absolute value of the entries:

the norm $\|A\|_{2,1}$ is the sum of $l_{2}$ norms of the columns of $A$ and $\|A\|_{2,\infty}$ is the largest $l_{2}$ norm of the columns of $A$:

For the columnwise sparse matrix $S_{0}$, we define

Let $|A|$ denote the matrix whose entries are the absolute values of the entries of matrix $A$. The norm $\mathcal{R}(\cdot)$ is called absolute if it depends only on the absolute values of the entries of $A$:

For instance, the $l_{p}$-norm and the $\|\cdot\|_{2,1}$-norm are absolute. We call $\mathcal{R}(\cdot)$ monotonic if $|A|\leq|B|$ implies $\mathcal{R}(A)\leq\mathcal{R}(B)$. Here and below, the inequalities between matrices are understood as entry-wise inequalities. Any absolute norm is monotonic and vice versa (see, e.g., ).

We set $d=m_{1}+m_{2}$, $m=m_{1}\wedge m_{2}$, and $M=m_{1}\vee m_{2}$.

Let $\{\epsilon_{i}\}_{i=1}^{n}$ be a sequence of i.i.d. Rademacher random variables. We define the following random variables called the stochastic terms:

We denote by $r$ the rank of matrix $L_{0}$.

We use the generic symbol $C$ for positive constants that do not depend on $n,m_{1},m_{2},r,s$ and can take different values at different appearances.

2 Convex relaxation for robust matrix completion

For the usual matrix completion, i.e., when the corruption matrix $S_{0}=\bf{0}$, one of the most popular methods of solving the problem is based on constrained nuclear norm minimization. For example, the following constrained matrix Lasso estimator is introduced in :

where $\lambda>0$ is a regularization parameter and $\mathbf{a}$ is an upper bound on $\left\|L_{0}\right\|_{\infty}$.

To account for the presence of non-zero corruptions $S_{0}$, we introduce an additional norm-based penalty that should be chosen depending on the structure of $S_{0}$. We consider the following estimator $(\hat{L},\hat{S})$ of the pair $(L_{0},S_{0})$:

Here $\lambda_{1}>0$ and $\lambda_{2}>0$ are regularization parameters and $\mathbf{a}$ is an upper bound on $\left\|L_{0}\right\|_{\infty}$ and $\left\|S_{0}\right\|_{\infty}$. Note that this definition and all the proofs can be easily adapted to the setting with two different upper bounds for $\left\|L_{0}\right\|_{\infty}$ and $\left\|S_{0}\right\|_{\infty}$ as it can be the case in some applications. Thus, the results of the paper extend to this case as well.

For the following two key examples of sparsity structure of $S_{0}$, we consider specific regularizers $\mathcal{R}$.

Example 1. Suppose that $S_{0}$ is columnwise sparse, that is, it has a small number $s<m_{2}$ of non-zero columns. We use the $\|\cdot\|_{2,1}$-norm regularizer for such a sparsity structure: $\mathcal{R}(S)=\|S\|_{2,1}.$ The associated dual norm is $\mathcal{R}^{*}(S)=\|S\|_{2,\infty}$.

Example 2. Suppose now that $S_{0}$ is entrywise sparse, that is, that it has $s\ll m_{1}m_{2}$ non-zero entries. The usual choice of regularizer for such a sparsity structure is the $l_{1}$ norm: $\mathcal{R}(S)=\|S\|_{1}$. The associated dual norm is $\mathcal{R}^{*}(S)=\|S\|_{\infty}$.

For instance, the $\|\cdot\|_{2,1}$-norm is decomposable with respect to any set $I$ such that

where $J\subset\{1,\dots,m_{2}\}$. The usual $l_{1}$ norm is decomposable with respect to any subset of indices $I$.

3 Assumptions on the sampling scheme and on the noise

In the literature on the usual matrix completion ($S_{0}=\bf{0}$), it is commonly assumed that the observations $X_{i}$ are i.i.d. For robust matrix completion, it is more realistic to assume the presence of two subsets in the observed $X_{i}$. The first subset $\{X_{i},\,i\in\Omega\}$ is a collection of i.i.d. random matrices with some unknown distribution on

These $X_{i}$’s are of the same type as in the usual matrix completion. They are the $X$-components of non-corrupted observations (recall that the entries of $S_{0}$ corresponding to indices in $\mathcal{I}$ are equal to zero). On this non-corrupted part of observations, we require some assumptions on the sampling distribution (see Assumptions 1, 2, 5, and 9 below).

There exists a positive constant $\mu\geq 1$ such that, for any $(j,k)\in\mathcal{I}$,

Denote by $\pi_{\cdot k}=\underset{j=1}{\overset{m_{1}}{\Sigma}}\pi_{jk}$ the probability to observe an element from the $k$-th column and by $\pi_{j\cdot}=\underset{k=1}{\overset{m_{2}}{\Sigma}}\pi_{jk}$ the probability to observe an element from the $j$-th row. The following assumption requires that no column and no row is sampled with too high probability.

There exists a positive constant $L\geq 1$ such that

This assumption will be used in Theorem 1 below. In Sections 4 and 5, we apply Theorem 1 to the particular cases of columnwise sparse and entrywise sparse corruptions. There, we will need more restrictive assumptions on the sampling distribution (see Assumptions 5 and 9).

We assume below that the noise variables $\xi_{i}$ are sub-gaussian:

There exist positive constants $\sigma$ and $c_{1}$ such that

Upper bounds for general regularizers

In this section we state our main result which applies to a general convex program (5) where $\mathcal{R}$ is an absolute norm and a decomposable regularizer. In the next sections, we consider in detail two particular choices, $\mathcal{R}(\cdot)=\|\cdot\|_{1}$ and $\mathcal{R}(\cdot)=\|\cdot\|_{2,1}$. Introduce the notation:

Let $\mathcal{R}$ be an absolute norm and a decomposable regularizer. Assume that $\left\|L_{0}\right\|_{\infty}\leq\mathbf{a}$, $\left\|S_{0}\right\|_{\infty}\leq\mathbf{a}$ for some constant $\mathbf{a}$ and let Assumptions 1 - 3 be satisfied. Let $\lambda_{1}>4\left\|\Sigma\right\|$, and $\lambda_{2}\geq 4\left(\mathcal{R}^{*}(\Sigma)+2\mathbf{a}\mathcal{R}^{*}(W)\right)$. Then, with probability at least $1-4.5\,d^{-1}$,

where $C$ is an absolute constant. Moreover, with the same probability,

The term $\Psi_{1}$ in (11) corresponds to the estimation error associated with matrix completion of a rank $r$ matrix. The second and the third terms account for the error induced by corruptions. In the next two sections we apply Theorem 1 to the settings with the entrywise sparse and columnwise sparse corruption matrices $S_{0}$.

Columnwise sparse corruptions

In this section, we assume that that $S_{0}$ has at most $s$ non-zero columns, and $s\leq m_{2}/2$. We use here the $\|\cdot\|_{2,1}$-norm regularizer $\mathcal{R}$. Then, the convex program (5) takes form

Specializing Theorem 1 to this case yields the following corollary.

Assume that $\left\|L_{0}\right\|_{\infty}\leq\mathbf{a}$ and $\left\|S_{0}\right\|_{\infty}\leq\mathbf{a}$. Let the regularization parameters $(\lambda_{1},\lambda_{2})$ satisfy

Then, with probability at least $1-4.5\,d^{-1}$, for any solution $(\hat{L},\hat{S})$ of the convex program (13) with such regularization parameters $(\lambda_{1},\lambda_{2})$ we have

where $C$ is an absolute constant. Moreover, with the same probability,

In order to get a bound in a closed form, we need to obtain suitable upper bounds on the stochastic terms $\Sigma$, $\Sigma_{R}$ and $W$. We derive such bounds under an additional assumption on the column marginal sampling distribution. Set $\pi_{\cdot,k}^{(2)}=\sum_{j=1}^{m_{1}}\pi_{jk}^{2}$.

There exists a positive constant $\gamma\geq 1$ such that

This condition prevents the columns from being sampled with too high probability and guarantees that the non-corrupted observations are well spread out among the columns. Assumption 5 is clearly less restrictive than assuming that $\Pi$ is uniform as it was done in the previous work on noiseless robust matrix completion. In particular, Assumption 5 is satisfied when the distribution $\Pi$ is approximately uniform, i.e., when $\pi_{jk}\asymp\frac{1}{m_{1}(m_{2}-s)}$. Note that Assumption 5 implies the following milder condition on the marginal sampling distribution:

Condition (14) is sufficient to control $\|\Sigma\|_{2,\infty}$ and $\|\Sigma_{R}\|_{2,\infty}$ while to we need a stronger Assumption 5 to control $\|W\|_{2,\infty}$.

The following lemma gives the order of magnitude of the stochastic terms driving the rates of convergence.

Let the distribution $\Pi$ on $\mathcal{X}^{\prime}$ satisfy Assumptions 1, 2 and 5. Let also Assumption 3 hold. Assume that $N\leq m_{1}m_{2}$, $n\leq|\mathcal{I}|$, and $\log m_{2}\geq 1$. Then, there exists an absolute constant $C>0$ such that, for any $t>0$, the following bounds on the norms of the stochastic terms hold with probability at least $1-e^{-t}$, as well as the associated bounds in expectation.

Recall that $\ae=\frac{N}{n}\geq 1$. If $n\geq n^{*}$, using the bounds given by Lemma 6, we can chose the regularization parameters $\lambda_{1}$ and $\lambda_{2}$ in the following way:

where $C>0$ is a large enough numerical constant.

With this choice of the regularization parameters, Corollary 4 implies the following result.

Let the distribution $\Pi$ on $\mathcal{X}^{\prime}$ satisfy Assumptions 1, 2 and 5. Let Assumption 3 hold and $\left\|L_{0}\right\|_{\infty}\leq\mathbf{a}$, $\left\|S_{0}\right\|_{\infty}\leq\mathbf{a}$. Assume that $N\leq m_{1}m_{2}$ and $n^{*}\leq n$. Then, with probability at least $1-6/d$ for any solution $(\hat{L},\hat{S})$ of the convex program (13) with the regularization parameters $(\lambda_{1},\lambda_{2})$ given by (16), we have

where $C_{\mu,\gamma,L}>0$ can depend only on $\mu,\gamma,L$. Moreover, with the same probability,

The estimator studied in for matrix decomposition problem is similar to our program (13). The difference between these estimators is that in (13) the minimization is over $\|\cdot\|_{\infty}$-balls while the program of uses the minimization over $\|\cdot\|_{2,\infty}$-balls and requires the knowledge of a bound on the norm $\|L_{0}\|_{2,\infty}$ of the unknown matrix $L_{0}$.

3. Suppose that the number of corrupted columns is small ($s\ll m_{2}$). Then, Corollary 7 guarantees, that the prediction error of our estimator is small whenever the number of non-corrupted observations $n$ satisfies the following condition

4. By changing the numerical constants, one can obtain that the upper bound (17) is valid with probability $1-6d^{-\alpha}$ for a given $\alpha\geq 1$.

Entrywise sparse corruptions

We assume now that $S_{0}$ has $s$ non-zero entries but they do not necessarily lay in a small subset of columns. We will also assume that $s\leq\frac{m_{1}m_{2}}{2}$. We use now the $l_{1}$-regularizer $\mathcal{R}$. Then the convex program (5) takes the form

Specializing Theorem 1 to this case yields the following corollary:

Assume that $\left\|L_{0}\right\|_{\infty}\leq\mathbf{a}$ and $\left\|S_{0}\right\|_{\infty}\leq\mathbf{a}$. Let the regularization parameters $(\lambda_{1},\lambda_{2})$ satisfy

Then, with probability at least $1-4.5\,d^{-1}$, for any solution $(\hat{L},\hat{S})$ of the convex program (19) with such regularization parameters $(\lambda_{1},\lambda_{2})$ we have

where $C$ is an absolute constant. Moreover, with the same probability,

In order to get a bound in a closed form we need to obtain suitable upper bounds on the stochastic terms $\Sigma,\Sigma_{R}$ and $W$. We provide such bounds under the following additional assumption on the sampling distribution.

There exists a positive constant $\gamma\geq 1$ such that

This assumption prevents any entry from being sampled too often and guarantees that the observations are well spread out over the non-corrupted entries. Assumptions 1 and 9 imply that the sampling distribution $\Pi$ is approximately uniform in the sense that $\pi_{jk}\asymp\frac{1}{|\mathcal{I}|}$. In particular, since $|\mathcal{I}|\leq\frac{m_{1}m_{2}}{2}$, Assumption 9 implies Assumption 2 for $L=2\mu_{1}$.

Let the distribution $\Pi$ on $\mathcal{X}^{\prime}$ satisfy Assumptions 1, and 9. Let also Assumption 3 hold. Then, there exists an absolute constant $C>0$ such that, for any $t>0$, the following bounds on the norms of the stochastic terms hold with probability at least $1-e^{-t}$, as well as the associated bounds in expectation.

Using Lemma 6(i), and Lemma 10, under the conditions

we can choose the regularization parameters $\lambda_{1}$ and $\lambda_{2}$ in the following way:

With this choice of the regularization parameters, Corollary 8 and Lemma 10 imply the following result.

Let the distribution $\Pi$ on $\mathcal{X}^{\prime}$ satisfy Assumptions 1, and 9. Let Assumption 3 hold and $\left\|L_{0}\right\|_{\infty}\leq\mathbf{a}$, $\left\|S_{0}\right\|_{\infty}\leq\mathbf{a}$. Assume that $N\leq m_{1}m_{2}$ and that condition (20) holds. Then, with probability at least $1-6/d$ for any solution $(\hat{L},\hat{S})$ of the convex program (19) with the regularization parameters $(\lambda_{1},\lambda_{2})$ given by (21), we have

where $C_{\mu,\mu_{1}}>0$ can depend only on $\mu$ and $\mu_{1}$. Moreover, with the same probability

2. If $s\ll n<m_{1}m_{2}$, the bound (22) implies that one can estimate a low-rank matrix from a nearly minimal number of observations, even when a part of the observations has been corrupted.

Minimax lower bounds

The following theorem gives a lower bound on the estimation risk in the case of columnwise sparsity.

We have the following theorem for the lower bound in the case of entrywise sparsity.

Appendix A Proofs of Theorem 1 and of Corollary 7

The proofs of the upper bounds have similarities with the methods developed in for noisy matrix completion but the presence of corruptions in our setting requires a new approach, in particular, for proving ”restricted strong convexity property” (Lemma 15) which is the main difficulty in the proof.

and our goal is to bound from above the Frobenius norms $\|L_{0}-\hat{L}\|_{2}^{2}$ and $\|S_{0}-\hat{S}\|_{2}^{2}$.

1) Set $\mathcal{F}(L,S)=\frac{1}{N}\sum^{N}_{i=1}\left(Y_{i}-\left\langle X_{i},L+S\right\rangle\right)^{2}+\lambda_{1}\|L\|_{*}+\lambda_{2}\mathcal{R}(S)$, $\Delta L=L_{0}-\hat{L}$ and $\Delta S=S_{0}-\hat{S}$. Using the inequality $\mathcal{F}(\hat{L},\hat{S})\leq\mathcal{F}(L_{0},S_{0})$ and (1) we get

where $\Sigma=\frac{1}{N}\sum_{i\in\Omega}\xi_{i}X_{i}$ and we have used the equality $\left\langle\Sigma,\Delta S\right\rangle=\left\langle\Sigma,\Delta S_{\mathcal{I}}\right\rangle$. We now estimate each of the three terms on the right hand side of (25) separately. This will be done on the random event

We start by estimating $\mathbf{I}$. On the event $\mathcal{U}$, we get

By definition of $\mathbf{P}_{L_{0}}^{\bot}$, for any matrix $B$ the singular vectors of $\mathbf{P}_{L_{0}}^{\bot}(B)$ are orthogonal to the space spanned by the singular vectors of $L_{0}$. This implies that $\left\|L_{0}+\mathbf{P}_{L_{0}}^{\bot}(\Delta L)\right\|_{*}=\left\|L_{0}\right\|_{*}+\left\|\mathbf{P}_{L_{0}}^{\bot}(\Delta L)\right\|_{*}$. Thus,

Using (28) and the duality between the nuclear and the operator norms, we obtain

The assumption that $\lambda_{1}\geq 4\|\Sigma\|$ and the triangle inequality imply

where $r={\rm rank}(L_{0})$ and we have used that ${\rm rank}(\mathbf{P}_{L_{0}}(\Delta L))\leq 2\,{\rm rank}(L_{0})$.

For the third term in (25), we use the duality between the $\mathcal{R}$ and $\mathcal{R}^{*}$, and the identity $\Delta S_{\mathcal{I}}=-\hat{S}_{\mathcal{I}}$:

This and the assumption that $\lambda_{2}\geq 4\mathcal{R}^{*}(\Sigma)$ imply

Plugging (29), (30) and (27) in (25) we get that, on the event $\mathcal{U}$,

2) Second, we will show that a kind of restricted strong convexity holds for the random sampling operator given by $(X_{i})$ on a suitable subset of matrices. In words, we prove that the observation operator captures a substantial component of any pair of matrices $L,S$ belonging to a properly chosen constrained set (cf. Lemma 15(ii) below for the exact statement). This will imply that, with high probability,

with an appropriate residual $\mathcal{E}$, whenever we prove that $(\Delta L,\Delta S)$ belongs to the constrained set. This will be a substantial element of the remaining part of the proof. The result of the theorem will then be deduced by combining (31) and (32).

We start by defining our constrained set. For positive constants $\delta_{1}$ and $\delta_{2}$, we first introduce the following set of matrices where $\Delta S$ should lie:

The constants $\delta_{1}$ and $\delta_{2}$ define the constraints on the $L_{2}(\Pi)$-norm and on the sparsity of the component $S$. The error term $\mathcal{E}$ in (32) depends on $\delta_{1}$ and $\delta_{2}$. We will specify the suitable values of $\delta_{1}$ and $\delta_{2}$ for the matrix $\Delta S$ later. Next, we define the following set of pairs of matrices:

where $\kappa$ and $\tau<m_{1}\wedge m_{2}$ are some positive constants. This will be used for $A=\Delta L$ and $B=\Delta S$. If the $L_{2}(\Pi)$-norm of the sum of two matrices is too small, the right hand side of (32) is negative. The first inequality in the definition of $\mathcal{D}(\tau,\kappa)$ prevents from this. Condition $\left\|A\right\|_{*}\leq\sqrt{\tau}\left\|A_{\mathcal{I}}\right\|_{2}+\kappa$ is a relaxed form of the condition $\left\|A\right\|_{*}\leq\sqrt{\tau}\left\|A\right\|_{2}$ satisfied by matrices with rank $\tau$. We will show that, with high probability, the matrix $\Delta L$ satisfies this condition with $\tau=C\,{\rm rank}(L_{0})$ and a small $\kappa$. To prove it, we need the bound $\mathcal{R}(B)\leq\delta_{2}$ on the corrupted part.

Finally, define our constrained set as the intersection

We now return to the proof of the theorem. To prove (11), we bound separately the norms $\left\|\Delta L\right\|_{2}$ and $\left\|\Delta S\right\|_{2}$. Note that

In view of these inequalities, it is enough to bound the quantities $\|\Delta S_{\mathcal{I}}\|_{L_{2}(\Pi)}^{2}$ and $\|\Delta L_{\mathcal{I}}\|_{2}^{2}$. A bound on $\|\Delta S_{\mathcal{I}}\|_{L_{2}(\Pi)}^{2}$ with the rate as claimed in (11) is given in Lemma 14 below. In order to bound $\|\Delta L_{\mathcal{I}}\|_{L_{2}(\Pi)}^{2}$ (or $\|\Delta L_{\mathcal{I}}\|_{2}^{2}$ according to cases), we will need the following argument.

Case 1: Suppose that $\left\|\Delta L+\Delta S\right\|_{L_{2}(\Pi)}^{2}<16\mathbf{a}^{2}\sqrt{\dfrac{64\,\log(d)}{\log\left(6/5\right)\,n}}$. Then a straightforward inequality

together with Lemma 14 below implies that, with probability at least $1-2.5/d$,

Note also that $\Psi_{4}\leq C(\Psi_{1}+\Psi_{2}+\Psi_{3})$. In view of (34), (36) and of fact that $|\mathcal{I}|\leq m_{1}m_{2}$, the bound on $\|\Delta L\|_{2}^{2}$ stated in the theorem holds with probability at least $1-2.5/d$.

Lemma 13 below and (27) imply that, on the event $\mathcal{U}$,

Lemma 14 yields that, with probability at least $1-2.5\,d^{-1}$,

Therefore, we can apply Lemma 15(ii). From Lemma 15(ii) and (31) we obtain that, with probability at least $1-4.5\,d^{-1}$,

Using an elementary argument and then (34) we find

Using again (35), Lemma 14, (9) and the bound $|\mathcal{I}|\leq m_{1}m_{2}$ we obtain

In view of (40) and (34), $\|\Delta L\|_{2}^{2}$ is bounded by the right hand side of (11) with probability at least $1-4.5\,d^{-1}$. Finally, inequality (12) follows from Lemma 14, (9) and the identity $\Delta S_{\mathcal{I}}=-\hat{S}_{\mathcal{I}}$.

Assume that $\lambda_{2}\geq 4\left(\mathcal{R}^{*}(\Sigma)+2\mathbf{a}\mathcal{R}^{*}(W)\right)$. Then, we have

Let $\partial\|\cdot\|_{*}$, and $\partial\mathcal{R}$ denote the subdifferentials of $\|\cdot\|_{*}$ and of $\mathcal{R}$, respectively. By the standard condition for optimality over a convex set (see , Chapter 4, Section 2, Corollary 6), we have

for all feasible pairs $(L,S)$. In particular, for $(\hat{L},S_{0})$ we obtain

Using the elementary inequality $2ab\leq a^{2}+b^{2}$ and the bound $\|\Delta L\|_{\infty}\leq 2\mathbf{a}$ we find

where $W=\frac{1}{N}\sum_{i\in\Omega}X_{i}$. On the other hand, the convexity of $\mathcal{R}(\cdot)$ and the definition of subdifferential imply

Next, the decomposability of $\mathcal{R}(\cdot)$, the identity $(S_{0})_{\mathcal{I}}=0$ and the triangle inequality yield

Since $\lambda_{2}\geq 4\left(2\mathbf{a}\mathcal{R}^{*}(W)+\mathcal{R}^{*}(\Sigma)\right)$ the last two displays imply

Suppose that $\lambda_{1}\geq 4\|\Sigma\|$ and $\lambda_{2}\geq 4\mathcal{R}^{*}(\Sigma)$. Then,

Using (41) for $(L,S)=(L_{0},S_{0})$ we obtain

The convexity of $\|\cdot\|_{*}$ and of $\mathcal{R}(\cdot)$ and the definition of the subdifferential imply

Using the conditions $\lambda_{1}\geq 4\|\Sigma\|$, $\lambda_{2}\geq 4\mathcal{R}^{*}(\Sigma)$, the triangle inequality and (28) we get

Let $n>m_{1}$ and $\lambda_{2}\geq 4\left(\mathcal{R}^{*}(\Sigma)+2\mathbf{a}\mathcal{R}^{*}(W)\right)$. Suppose that the distribution $\Pi$ on $\mathcal{X}^{\prime}$ satisfies Assumptions 1 and 2. Let $\left\|S_{0}\right\|_{\infty}\leq\mathbf{a}$ for some constant $\mathbf{a}$ and let Assumption 3 be satisfied. Then, with probability at least $1-2.5\,d^{-1}$,

Using the inequality $\mathcal{F}(\hat{L},\hat{S})\leq\mathcal{F}(\hat{L},S_{0})$ and (1) we obtain

From Lemma 18 and the duality between $\mathcal{R}$ and $\mathcal{R}^{*}$ we obtain

Since here $\Delta S_{\mathcal{I}}=-\hat{S}_{\mathcal{I}}$ and $\lambda_{2}\geq 4\left(\mathcal{R}^{*}(\Sigma)+2\mathbf{a}\mathcal{R}^{*}(W)\right)$ it follows that

Now, Lemma 12 and the bound $\|\Delta S\|_{\infty}\leq 2\mathbf{a}$ imply that, on the event $\mathcal{U}$ defined in (26),

Thus, (48) is proved. To prove (47), consider the following two cases.

Case I: $\|\Delta S\|^{2}_{L_{2}(\Pi)}<4\mathbf{a}^{2}\sqrt{\frac{64\log(d)}{\log(6/5)\,n}}$. Then (47) holds trivially.

Case II: $\|\Delta S\|^{2}_{L_{2}(\Pi)}\geq 4\mathbf{a}^{2}\sqrt{\frac{64\log(d)}{\log(6/5)\,n}}$. Then inequality (50) and the bound $\|\Delta S\|_{\infty}\leq 2\mathbf{a}$ imply that, on the event $\mathcal{U}$,

where, for any $\delta>0$, the set $\mathcal{C}(\delta)$ is defined as:

Let the distribution $\Pi$ on $\mathcal{X}^{\prime}$ satisfy Assumptions 1 and 2. Let $\delta,\delta_{1},\delta_{2},\tau$, and $\kappa$ be positive constants. Then, the following properties hold.

With probability at least $1-\dfrac{2}{d}$,

With probability at least $1-\dfrac{2}{d}$,

We give a unified proof of (i) and (ii). Let $A=S$ for (i) and $A=L+S$ for (ii). Set

To prove the lemma it is enough to show that the probability of the random event

where we have used the inequality $e^{x}\geq x$. We finally obtain, for $\nu=\sqrt{\dfrac{64\,\log(d)}{\log\left(6/5\right)\,n}}$,

Let the distribution $\Pi$ on $\mathcal{X}^{\prime}$ satisfy Assumptions 1 and 2. Then,

We follow a standard approach: first we show that $Z_{T}$ concentrates around its expectation and then we bound from above the expectation. Since $\left\|A\right\|_{\infty}\leq 1$ for all $A\in\mathcal{C}^{\prime}(T)$, we have $\left|\left\langle X_{i},A\right\rangle\right|\leq 1$. We use first a Talagrand type concentration inequality, cf. [4, Theorem 14.2], implying that

where $\{\epsilon_{i}\}_{i=1}^{n}$ is an i.i.d. Rademacher sequence. Then, the contraction inequality (see e.g. ) yields

Case I: $A\in\mathcal{C}(\delta)$ and $\left\|A\right\|_{L_{2}(\Pi)}^{2}\leq T$. By the definition of $\mathcal{C}(\delta)$ we have $\mathcal{R}(A)\leq\delta.$ Thus, by the duality between $\mathcal{R}$ and $\mathcal{R}^{*}$,

This and the concentration inequality (53) imply

Case II: $A=L+S$ where $(L,S)\in\mathcal{D}(\tau,\kappa)$, $S\in\mathcal{B}(\delta_{1},\delta_{2})$, and $\|L+S\|^{2}_{L_{2}(\Pi)}\leq T$. Then, by the definition of $\mathcal{B}(\delta_{1},\delta_{2})$, we have $\mathcal{R}(S)\leq\delta_{2}.$ On the other hand, the definition of $\mathcal{D}(\tau,\kappa)$ yields

Combining this bound with the following elementary inequalities:

and using the concentration bound (53) we obtain

A.2 Proof of Corollary 7

With $\lambda_{1}$ and $\lambda_{2}$ given by (16) we obtain

Appendix B Proof of Theorems 2 and 3

Note that the §assumption $\ae\leq 1+s/m_{2}$ implies that

Assume w.l.o.g. that $m_{1}\geq m_{2}$. For a $\gamma\leq 1$, define

and consider the associated set of block matrices

where $O$ denotes the $m_{1}\times(m_{2}-r\lfloor m_{2}/(2r)\rfloor)$ zero matrix, and $\lfloor x\rfloor$ is the integer part of $x$.

By construction, any element of $\mathcal{A}$ as well as the difference of any two elements of $\mathcal{A}$ can be decomposed into a low rank component $L$ of rank at most $r$ and a group sparse component $S$ with at most $s$ nonzero columns. In addition, the entries of any matrix in $\mathcal{A}$ take values in $[0,a]$. Thus, $\mathcal{A}\subset{\cal A}_{GS}(r,s,\mathbf{a})$.

where we have used Assumption 9. From (57) we deduce that the condition

is satisfied if $\gamma>0$ is chosen as a sufficiently small numerical constant. In view of (56) and (58), the application of Theorem 2.5 in implies

for some absolute constants $\beta\in(0,1)$.

which implies that condition (58) is satisfied if $\gamma>0$ is chosen small enough. Thus, applying Theorem 2.5 in we get

for some absolute constant $\beta\in(0,1)$. Theorem 2 follows from inequalities (55), (59) and (61).

Appendix C Proof of Lemma 6

Part (i) of Lemma 6 is proved in Lemmas 5 and 6 in .

Proof of (ii). For the sake of brevity, we set $X_{i}(j,k)=\langle X_{i},e_{j}(m_{1})e_{k}(m_{2})^{\top}\rangle$. By definition of $\Sigma$ and $\|\cdot\|_{2,\infty}$, we have

We freeze the $X_{i}$ and we apply the version of Hanson–Wright inequality in to get that there exists a numerical constant $C$ such that with probability at least $1-e^{-t}$

where we have used the Cauchy - Schwarz inequality in the first line and the relation $X^{2}_{i}(j,k)=X_{i}(j,k)$.

Note that $Z_{i}(k):=\sum_{j=1}^{m_{1}}X_{i}(j,k)$ follows a Bernoulli distribution with parameter $\pi_{\cdot k}$ and consequently $Z(k)=\sum_{i\in\Omega}Z_{i}(k)$ follows a Binomial distribution $B(|\Omega|,\pi_{\cdot k})$. We apply Bernstein’s inequality (see, e.g., , page 486) to get that, for any $t>0$,

Consequently, we get with probability at least $1-2e^{-t}$ that

and, using $\|A_{k}\|\leq\|A_{k}\|_{2}$, that

Combining the last three displays with (63) we get, up to a rescaling of the constants, with probability at least $1-e^{-t}$ that

Replacing $t$ by $t+\log m_{2}$ in the above display and using the union bound gives that, with probability at least $1-e^{-t}$,

Assuming that $\log m_{2}\geq 1$ we get with probability at least $1-e^{-t}$ that

Using (14), we get that there exists a numerical constant $C>0$ such with probability at least $1-e^{-t}$

Proof of (iii). We follow the same lines as in the proof of part (ii) above. The only difference is to replace $\xi_{i}$ by $\epsilon_{i}$, $\sigma$ by $1$ and $N$ by $n$.

Proof of (iv). We need to establish the bound on

The first term on the right hand side of the last display can be written as

Using the concentration bound on $Z(k)$ in the proof of part (ii) above, we get that, with probability at least $1-e^{-t}$,

is a U-statistic of order 2. We use now a Bernstein-type concentration inequality for U-statistics. To this end, we set $X_{i}(\cdot,k)=(X_{i}(1,k),\cdots,X_{i}(m_{1},k))^{\top}$ and

We will need the following quantities to control the tail behavior of $U_{2}$

We now evaluate the above quantities in our particular setting. It is not hard to see that $\mathbf{A}=\max\{\pi_{\cdot k}^{(2)}\,,\,1-\pi_{\cdot k}^{(2)}\}\leq 1$ where $\pi_{\cdot k}^{(2)}=\sum_{j=1}^{m_{1}}\pi_{jk}^{2}$. We also have that

where we have used in the second line that $\langle X_{i_{1}}(\cdot,k),X_{i_{2}}^{\prime}(\cdot,k)\rangle^{2}=\langle X_{i_{1}}(\cdot,k),X_{i_{2}}^{\prime}(\cdot,k)\rangle$ since $\langle X_{i_{1}}(\cdot,k),X_{i_{2}}^{\prime}(\cdot,k)\rangle$ takes values in $\{0,1\}$.

We now derive a bound on $\mathbf{D}$. By Jensen’s inequality, we get

Finally, we get a bound on $\mathbf{B}$. Set $\pi_{0,k}=1-\pi_{\cdot,k}$. Note first that

Set now $U_{2}=\sum_{i_{1}\neq i_{2}}h(X_{i_{1}}(\cdot,k),X_{i_{2}}(\cdot,k)).$ We apply a decoupling argument (See for instance Theorem 3.4.1 page 125 in ) to get that there exists a constant $C>0$, such that for any $u>0$

where $X_{i}^{\prime}(\cdot,k)$ is independent of $X_{i}(\cdot,k)$ and has the same distribution as $X_{i}(\cdot,k)$. Next, Theorem 3.3 in gives that, for any $u>0$,

for some absolute constant $C>0$. Combining the last display with our bounds on $\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}$, we get that for any $t>0$, with probability at least $1-2e^{-t}$,

where $C>0$ is a numerical constant. Combining the last display with (64) we get that, for any $t>0$ with probability at least $1-3e^{-t}$,

Set $\pi_{\max}=\max_{1\leq k\leq m_{2}}\{\pi_{\cdot k}\}$ and $\pi^{(2)}_{\max}=\max_{1\leq k\leq m_{2}}\{\pi_{\cdot k}^{(2)}\}$. Using the union bound and up to a rescaling of the constants, we get that, with probability at least $1-e^{t}$,

Recall that $|\Omega|=n$ and $\ae=N/n$. Assumption 5 and the fact that $n\leq|\mathcal{I}|$ imply that there exists a numerical constant $C>0$ such that, with probability at least $1-e^{-t}$,

Appendix D Proof of Lemma 10

With the notation $X_{i}(j,k)=\langle X_{i},e_{j}(m_{1})e_{k}(m_{2})^{\top}\rangle$ we have

Thus, we can apply Bernstein’s inequality (see, e.g. , page 486), which yields

for any fixed $(j,k)$. Replacing here $t$ by $t+\log(m_{1}m_{2})$ and using the union bound we obtain

Appendix E Technical Lemmas

Let $Y$ be a non-negative random variable. Let there exist $A\geq 0$, and $a_{j}>0$, $\alpha_{j}>0$ for $1\leq j\leq m$, such that

where $\Gamma(\cdot)$ is the Gamma function.

Using the change of variable $u=\sum_{j=1}^{m}a_{j}v^{\alpha_{j}}$ we get

Assume that $\mathcal{R}$ is an absolute norm. Then

where $W=\frac{1}{N}\sum_{i\in\Omega}X_{i}$.

In view of the definition of $\mathcal{R}^{*}$,

where we have used the inequalities $\left\langle X_{i},\Delta L\right\rangle\leq\|\Delta L\|_{\infty}\leq 2\mathbf{a}$, and the fact that $\mathcal{R}$ is an absolute norm. ∎

Acknowledgement

The work of O. Klopp was conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01). The work of K. Lounici was supported in part by Simons Grant 315477 and by NSF Career Grant DMS-1454515. The work of A.B.Tsybakov was supported by GENES and by the French National Research Agency (ANR) under the grants IPANEMA (ANR-13-BSH1-0004-02), Labex ECODEC (ANR - 11-LABEX-0047), ANR -11- IDEX-0003-02, and by the ”Chaire Economie et Gestion des Nouvelles Données”, under the auspices of Institut Louis Bachelier, Havas-Media and Paris-Dauphine.

References