Sharp Restricted Isometry Property Bounds for Low-rank Matrix Recovery Problems with Corrupted Measurements

Low-rank matrix recovery problems arise in various applications, such as matrix completion (Candès and Recht 2009; Recht, Fazel, and Parrilo 2010), phase synchronization/retrieval (Singer 2011; Boumal 2016; Shechtman et al. 2015), robust PCA (Ge, Jin, and Zheng 2017), and several others (Chen and Chi 2018; Chi, Lu, and Chen 2019). In this paper, we study a class of low-rank matrix recovery problems, where the goal is to recover a symmetric and positive semidefinite ground truth matrix $M^{*}$ with $\operatorname{rank}(M^{*})=r$ from certain linear measurements corrupted by noise. This problem can be formulated as the following optimization problem:

Here, $\mathcal{A}:\mathbb{R}^{n\times n}\to\mathbb{R}^{m}$ is a linear operator whose action on a matrix $M$ is given by

where $A_{1},\dots,A_{m}\in\mathbb{R}^{n\times n}$ are called sensing matrices. In addition, $b=\mathcal{A}(M^{*})$ represents the perfect measurement on the ground truth $M^{*}$ and $w$ comes from an arbitrary probability distribution. Note that only the noisy measurement $b-w$ is available to the user, and indeed $b$ is unknown. In other words, from a problem-solving perspective, the random variable $w$ is hidden to the user, and it is explicitly modeled here only for the sake of analysis.

The modeling of the noise in this problem is critical for many practical applications in which the influence of the noise cannot be ignored. For example, state estimation is a major data analytic problem for the operation of power grids, which can be modeled as matrix sensing (Jin et al. 2019b). In this problem, each measurement comes from a physical device, and the noise in our formulation models not only the sensory noise but also mismatch between the true model of the system and the one used by a power operator, changes in the measurements due to cyber-attacks, mechanical faults, etc. In other words, by a noisy problem we mean a learning problem where a part of the data is wrong for various reasons, and as with this case, it is impossible to assume that the measurements are noise-free.

Although it may be possible to solve the problem (1) based on convex relaxations (Candès and Recht 2009; Recht, Fazel, and Parrilo 2010; Candès and Tao 2010), the computational complexity associated with solving a semidefinite program presents a major challenge for large-scale problems. A more scalable approach is to use the Burer–Monteiro factorization (Burer and Monteiro 2003) by expressing $M$ as $XX^{T}$ with $X\in\mathbb{R}^{n\times r}$, which leads to the following equivalent formulation of the original problem (1):

The unconstrained problem (2) is often solved by local search methods such as gradient descent. Since the objective function $f(X)$ in (2) is non-convex, local search algorithms may converge to a local minimizer, leading to a suboptimal or plainly wrong solution. Hence, it is desirable to provide guarantees on the maximum distance between these local minimizers and the ground truth $M^{*}$. This problem will be addressed in this paper.

We first present the global guarantee on the local minimizers of the problem (2). To simplify the notation, we use a matrix representation of the measurement operator $\mathcal{A}$ as follows:

Then, $\mathbf{A}\operatorname{vec}(M)=\mathcal{A}(M)$ for every matrix $M\in\mathbb{R}^{n\times n}$.

Note that two upper bounds on the distance $\lVert\hat{X}\hat{X}^{T}-M^{*}\rVert_{F}$ can be obtained for any local minimizer $\hat{X}$ by solving the two quadratic-like inequalities (3a) and (3b), and the larger bound needs to be used because only one of the two inequalities is guaranteed to hold. The explicit upper bound solved from Theorem 2 is given in Appendix A. The reason for the existence of two inequalities in Theorem 2 is the split of its proof into two cases. The first case is when the $r$-th smallest singular value of $\hat{X}$ is small, and the second case is the opposite, which are respectively handled by Lemma 2 and Lemma 3.

The reason for the existence of two inequalities in Theorem 2 is the split of its proof into two cases. The first case is when the $r$-th smallest singular value of $\hat{X}$ is small, and the second case is the opposite, which are respectively handled by Lemma 2 and Lemma 3.

Theorem 2 is a major extension of the existing sharp result stating that the noiseless problem has no spurious local minima under the same assumption of the $\delta$-$\operatorname{RIP}_{2r}$ property with $\delta<1/2$. The reason is that in the case when the noise $w$ is equal to zero, one can choose an arbitrarily small $\epsilon$ in Theorem 2 to conclude from the inequalities (3a) and (3b) that $\hat{X}\hat{X}^{T}=M^{*}$ for every local minimizer $\hat{X}$. Moreover, when the RIP constant $\delta$ further decreases from $1/2$, the upper bound on $\lVert\hat{X}\hat{X}^{T}-M^{*}\rVert_{F}$ will also decrease, which means that a local minimizer found by local search methods will be closer to the ground truth $M^{*}$. This suggests that the RIP condition is able to not only guarantee the absence of spurious local minima as shown in the previous literature but also mitigate the influence of the noise in the measurements.

Compared with the existing results such as Theorem 1, our new result has two advantages. First, by improving the RIP constant from $1/10$ to $1/2$, one can apply the results on the location of spurious local minima to a much broader class of problems, which can often help reduce the number of measurements. For example, in the case when the measurements are given by random Gaussian matrices, it is proven in Candès and Recht (2009) that to achieve the $\delta$-$\operatorname{RIP}_{2r}$ property the minimum number of measurements needed is in the order of $O(1/\delta^{2})$. By improving the RIP constant in the bound, we can significantly reduce the number of measurements while still keeping the benign landscape. In applications such as learning for energy networks, there is a fundamental limit on the number of measurements that can be collected due to the physics of the problem (Jin et al. 2021). Finding a better bound on RIP helps with addressing the issues with the number of measurements needed to reliably solve the problem. Second, Theorem 1 is just about the probability of having all spurious solutions in a fixed ball around the ground truth of radius $O(\sigma_{w})$ instead of balls of arbitrary radii, and this fixed ball could be a large one depending on whether the noise level $\sigma_{w}$ is fixed or scales with the problem. On the other hand, in Theorem 2, we consider the probability $\mathbb{P}(\lVert\mathbf{A}^{T}w\rVert\leq\epsilon)$ for any arbitrary value of $\epsilon$. By having a flexible $\epsilon$, our work not only improves the RIP constant but also allows computing the probability of having all spurious solutions in any given ball.

In the special case of rank $r=1$, the conditions (3a) and (3b) in Theorem 2 can be substituted with a simpler condition as shown below. Its proof is highly similar to that of Lemma 3, and obviated in this paper for succinctness.

In the case when the RIP constant $\delta$ is not less than $1/2$, it is not possible to achieve a global guarantee similar to Theorem 2 or Theorem 3 since it is known that the problem may have a spurious solution even in the noiseless case. Instead, we turn to local guarantees by showing that every arbitrary local minimizer $\hat{X}$ of problem (2) is either close to the ground truth $M^{*}$ or far away from it in terms of the distance $\lVert\hat{X}\hat{X}^{T}-M^{*}\rVert_{F}$.

The upper bounds in (5) and (6) define an outer ball and an inner ball centered at the ground truth $M^{*}$. Theorem 4 states that there is no local minimizer in the ring between the two balls. As $\epsilon$ approaches zero, the inner ball shrinks to the ground truth. This theorem shows that bad local minimizers are located outside the outer ball. Note that the problem could be highly non-convex when $\delta$ is close to 1, while this theorem shows a benign landscape in a local neighborhood of the solution. Furthermore, all the theorems in this section are applicable to arbitrary noise models since they make no explicit use of the probability distribution of the noise. The only required information is the probability $\mathbb{P}(\lVert\mathbf{A}^{T}w\rVert\leq\epsilon)$, which can be computed or bounded when the probability distribution of the noise is given as illustrated in Section 4.

The results presented above are all about the location of the local minimizers. They do not directly lead to the global convergence of local search methods with a fast convergence rate. To provide performance guarantees for local search methods, the next theorem establishes a stronger property for the landscape of the noisy problem that is usually called the strict saddle property in the literature, which essentially says that all approximate second-order critical points are close to the ground truth.

Note that this property is not proven in the literature for $\delta<1/2$ even in the noiseless case, and thus our result generalizes the existing ones even in this scenario. On the other hand, it is proven by Jin et al. (2017) that the perturbed gradient descent method with an arbitrary initialization will find a solution $\hat{X}$ satisfying the requirements in Theorem 5 with a high probability in $O(\mathrm{poly}(1/\kappa))$ number of iterations. By Theorem 5, $\hat{X}\hat{X}^{T}$ will be close to the ground truth if $\epsilon$ and $\kappa$ are chosen to be relatively small.

Table 1 briefly summarizes our result compared with the existing literature.

Before presenting the proofs, we first compute the gradient and the Hessian of the objective function $f(\hat{X})$ of the problem (2):

where

and $\hat{\mathbf{X}}\in\mathbb{R}^{n^{2}\times nr}$ is the matrix satisfying

The first step in the proofs is to derive necessary conditions for a matrix $\hat{X}\in\mathbb{R}^{n\times r}$ to be an approximate second-order critical point, which depend on the linear operator $\mathcal{A}$, the noise $w\in\mathbb{R}^{m}$, the solution $\hat{X}$, and the parameter $\kappa$ characterizing how close $\hat{X}$ is to a true second-order critical point.

If $\hat{X}$ is a local minimizer of the problem (2), Lemma 1 shows that $\hat{X}$ satisfies the inequalities (8a) and (8b) with $\kappa=0$. Similarly, Theorem 2 can also be regarded as a special case of Theorem 5 with $\kappa=0$. The proofs of these two theorems consist of inspecting two cases. The following lemma deals with the first case in which $\hat{X}$ is an approximate second-order critical point with $\sigma_{r}(\hat{X})$ being close to zero.

The remaining case with

will be handled in the following lemma using a different method.

The proofs of both Lemma 3 and the local guarantee in Theorem 4 generalize the proof of the absence of spurious local minima for the noiseless problem in Zhang and Zhang (2020); Zhang (2021). Our innovation here is to develop new techniques to analyze approximate optimality conditions for the solutions because unlike the noiseless problem the local minimizers of the noisy one are only approximate second-order critical points of the distance function $\|\mathcal{A}(XX^{T})-b\|^{2}$. For a fixed solution $\hat{X}$ and noise $w$, one can find an operator $\hat{\mathcal{A}}$ satisfying the $\delta$-$\operatorname{RIP}_{2r}$ property with the smallest possible $\delta$ such that $\hat{X}$ and $\hat{\mathcal{A}}$ satisfy the necessary conditions stated in Lemma 1. Let $\delta^{*}(\hat{X})$ be the RIP constant of the found measurement operator $\hat{\mathcal{A}}$ in the worst-case scenario. Then, if $\hat{X}$ in Lemma 3 is a solution of the current problem with the linear operator $\mathcal{A}$ satisfying the $\delta$-$\operatorname{RIP}_{2r}$ property, it holds that $\delta\geq\delta^{*}(\hat{X})$, which can further lead to an upper bound on the distance $\lVert\hat{X}\hat{X}^{T}-M^{*}\rVert_{F}$.

To compute $\delta^{*}(\hat{X})$ defined above, let $q=\mathbf{A}^{T}w$ and solve the following optimization problem whose optimal value is $\delta^{*}(\hat{X})$:

Note that a matrix $\hat{\mathbf{H}}\in\mathbb{R}^{n^{2}\times n^{2}}$ is said to satisfy the $\delta$-$\operatorname{RIP}_{2r}$ property if

holds for every matrix $U\in\mathbb{R}^{n\times n}$ with $\operatorname{rank}(U)\leq 2r$ and $\mathbf{U}=\operatorname{vec}(U)$. Obviously, for a linear operator $\hat{\mathcal{A}}$, $\hat{\mathbf{H}}=\hat{\mathbf{A}}^{T}\hat{\mathbf{A}}$ satisfies the $\delta$-$\operatorname{RIP}_{2r}$ property if and only if $\hat{\mathbf{A}}$ satisfies the $\delta$-$\operatorname{RIP}_{2r}$ property.

However, since problem (10) is non-convex due to the RIP constraint, we instead solve the following convex reformulation:

Lemma 14 in Bi and Lavaei (2020) proves that problem (10) and problem (11) have the same optimal value. The remaining step in the proof of Lemma 3 is to solve the optimization problem (11) for given $\hat{X}$, $q$ and $\kappa$. The complete proof of Lemma 3 is lengthy and deferred to Appendix B. Finally, Theorem 2 and Theorem 5 are direct consequences of Lemma 2 and Lemma 3. The proof of Theorem 3 is very similar to that of Lemma 3 and is also given in Appendix B.

Now, we turn to the proof of the local guarantee in Theorem 4. The following existing result will be useful.