It has been observed that the performances of many high-dimensional estimation problems are universal with respect to underlying sensing (or design) matrices. Specifically, matrices with markedly different constructions seem to achieve identical performance if they share the same spectral distribution and have “generic” singular vectors. We prove this universality phenomenon for the case of convex regularized least squares (RLS) estimators under a linear regression model with additive Gaussian noise. Our main contributions are two-fold: (1) We introduce a notion of universality classes for sensing matrices, defined through a set of deterministic conditions that fix the spectrum of the sensing matrix and precisely capture the notion of generic singular vectors; (2) We show that for all sensing matrices that lie in the same universality class, the dynamics of the proximal gradient descent algorithm for solving the regression problem, as well as the performance of RLS estimators themselves (under additional strong convexity conditions) are asymptotically identical. In addition to including i.i.d. Gaussian and rotational invariant matrices as special cases, our universality class also contains highly structured, strongly dependent, and even (nearly) deterministic matrices. Examples of the latter include randomly signed versions of incoherent tight frames and randomly subsampled Hadamard transforms. As a consequence of this universality principle, the asymptotic performance of regularized linear regression on many structured matrices constructed with limited randomness can be characterized by using the rotationally invariant ensemble as an equivalent yet mathematically more tractable surrogate.

In the phase retrieval problem one seeks to recover an unknown$n$dimensional signal vector$\mathbf {x}$from$m$measurements of the form$y_{i} = |(\mathbf {A} \mathbf {x})_{i}|$, where$\mathbf {A}$denotes the sensing matrix. Many algorithms for this problem are based on approximate message passing. For these algorithms, it is known that if the sensing matrix$\mathbf {A}$is generated by sub-sampling$n$columns of a uniformly random (i.e., Haar distributed) orthogonal matrix, in the high dimensional asymptotic regime ($m,n \rightarrow \infty, n/m \rightarrow \kappa $), the dynamics of the algorithm are given by a deterministic recursion known as the state evolution. For a special class of linearized message-passing algorithms, we show that the state evolution is universal: it continues to hold even when$\mathbf {A}$is generated by randomly sub-sampling columns of the Hadamard-Walsh matrix, if the signal is drawn from a Gaussian prior.

In the Tensor PCA problem introduced by Richard and Montanari (2014), one is given a dataset consisting of $n$ samples $\mathbf{T}_{1:n}$ of i.i.d. Gaussian tensors of order $k$ with the promise that $\mathbb{E}\mathbf{T}_1$ is a rank-1 tensor and $\|\mathbb{E} \mathbf{T}_1\| = 1$. The goal is to estimate $\mathbb{E} \mathbf{T}_1$. This problem exhibits a large conjectured hard phase when $k>2$: When $d \lesssim n \ll d^{\frac{k}{2}}$ it is information theoretically possible to estimate $\mathbb{E} \mathbf{T}_1$, but no polynomial time estimator is known. We provide a sharp analysis of the optimal sample complexity in the Statistical Query (SQ) model and show that SQ algorithms with polynomial query complexity not only fail to solve Tensor PCA in the conjectured hard phase, but also have a strictly sub-optimal sample complexity compared to some polynomial time estimators such as the Richard-Montanari spectral estimator. Our analysis reveals that the optimal sample complexity in the SQ model depends on whether $\mathbb{E} \mathbf{T}_1$ is symmetric or not. For symmetric, even order tensors, we also isolate a sample size regime in which it is possible to test if $\mathbb{E} \mathbf{T}_1 = \mathbf{0}$ or $\mathbb{E}\mathbf{T}_1 \neq \mathbf{0}$ with polynomially many queries but not estimate $\mathbb{E}\mathbf{T}_1$. Our proofs rely on the Fourier analytic approach of Feldman, Perkins and Vempala (2018) to prove sharp SQ lower bounds.

Phase retrieval refers to the problem of recovering a signal$ {x}_{\star }\in \mathbb {C}^{n}$from its phaseless measurements$\text {y}_{\text {i}}=| {a}_{i}^{ \mathsf {H}} {x}_{\star }|$, where$\{ {a}_{\text {i}}\}_{\text {i}=1}^{ {m}}$are the measurement vectors. Spectral method is widely used for initialization in many phase retrieval algorithms. The quality of spectral initialization can have a major impact on the overall algorithm. In this paper, we focus on the model where$ {A}=[ {a}_{1},\ldots, {a}_{ {m}}]^{ \mathsf {H}}$has orthonormal columns, and study the spectral initialization under the asymptotic setting$ {m}, {n}\to \infty $with$ {m}/ {n}\to \delta \in (1,\infty)$. We use the expectation propagation framework to characterize the performance of spectral initialization for Haar distributed matrices. Our numerical results confirm that the predictions of the EP method are accurate for not-only Haar distributed matrices, but also for realistic Fourier based models (e.g. the coded diffraction model). The main findings of this paper are the following: 1) There exists a threshold on$\delta $(denoted as$\delta _{ \mathrm {weak}}$) below which the spectral method cannot produce a meaningful estimate. We show that$\delta _{ \mathrm {weak}}=2$for the column-orthonormal model. In contrast, previous results by Mondelli and Montanari show that$\delta _{ \mathrm {weak}}=1$for the i.i.d. Gaussian model. 2) The optimal design for the spectral method coincides with that for the i.i.d. Gaussian model, where the latter was recently introduced by Luo, Alghamdi and Lu.

Phase retrieval refers to algorithmic methods for recovering a signal from its phaseless measurements. There has been recent interest in understanding the performance of local search algorithms that work directly on the non-convex formulation of the problem. Due to the non-convexity of the problem, the success of these local search algorithms depends heavily on their starting points. The most widely used initialization scheme is the spectral method, in which the leading eigenvector of a data-dependent matrix is used as a starting point. Recently, the performance of the spectral initialization was characterized accurately for measurement matrices with independent and identically distributed entries. This paper aims to obtain the same level of knowledge for isotropically random column-orthogonal matrices, which are substantially better models for practical phase retrieval systems. Towards this goal, we consider the asymptotic setting in which the number of measurements m, and the dimension of the signal, n, diverge to infinity with m/n = δ ∈ (1, ∞), and obtain a simple expression for the overlap between the spectral estimator and the true signal vector.

We study information theoretic limits of recovering an unknown n dimensional, complex signal vector x*with unit norm from m magnitude-only measurements of the form yi= |(Ax*)i|2, i = 1, 2 ..., m, where A is the sensing matrix. This is known as the Phase Retrieval problem and models practical imaging systems where measuring the phase of the observations is difficult. Since in a number of applications, the sensing matrix has orthogonal columns, we model the sensing matrix as a subsampled Haar matrix formed by picking n columns of a uniformly random m X m unitary matrix. We study this problem in the high dimensional asymptotic regime, where m, n → ∞, while m/n → δ with δ being a fixed number, and show that if mn(1)) · n, then any estimator is asymptotically orthogonal to the true signal vector x*. This lower bound is sharp since when m > (2 + on(1)) · n, estimators that achieve a non trivial asymptotic correlation with the signal vector are known from previous works.

We study the problem of learning a real-valued function of correlated variables. Solving this problem is of interest since many classical learning results apply only in the case of learning functions of random variables that are independent. We show how to recover a high-dimensional, sparse monomial model from Gaussian examples with sample complexity that is poly-logarithmic in the total number of variables and polynomial in the number of relevant variables. Our algorithm is based on a transformation of the variables—taking their logarithm—followed by a sparse linear regression procedure, which is statistically and computationally efficient. While this transformation is commonly used in applied non-linear regression, its statistical guarantees have never been rigorously analyzed. We prove that the sparse regression procedure succeeds even in cases where the original features are highly correlated and fail to satisfy the standard assumptions required for sparse linear regression.

We consider regression problems where the response is a smooth but non-linear function of a $k$-dimensional projection of $p$ normally-distributed covariates, contaminated with additive Gaussian noise. The goal is to recover the range of the $k$-dimensional projection, i.e., the index space. This model is called the multi-index model, and the $k=1$ case is called the single-index model. For the single-index model, we characterize the population landscape of a natural semi-parametric maximum likelihood objective in terms of the link function and prove that it has no spurious local minima. We also propose and analyze an efficient iterative procedure that recovers the index space up to error $\epsilon$ using a sample size $\tilde{O}(p^{O(R^2/\mu)} + p/\epsilon^2)$, where $R$ and $\mu$, respectively, parameterize the smoothness of the link function and the signal strength. When a multi-index model is incorrectly specified as a single-index model, we prove that essentially the same procedure, with sample size $\tilde{O}(p^{O(kR^2/\mu)} + p/\epsilon^2)$, returns a vector that is $\epsilon$-close to being completely in the index space.