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The nonconvex formulation of matrix completion problem has received significant attention in recent years due to its affordable complexity compared to the convex formulation. Gradient descent (GD) is the simplest yet efficient baseline algorithm for solving nonconvex optimization problems. The success of GD has been witnessed in many different problems in both theory and practice when it is combined with random initialization. However, previous works on matrix completion require either careful initialization or regularizers to prove the convergence of GD. In this work, we study the rank-1 symmetric matrix completion and prove that GD converges to the ground truth when small random initialization is used. We show that in logarithmic amount of iterations, the trajectory enters the region where local convergence occurs. We provide an upper bound on the initialization size that is sufficient to guarantee the convergence and show that a larger initialization can be used as more samples are available. We observe that implicit regularization effect of GD plays a critical role in the analysis, and for the entire trajectory, it prevents each entry from becoming much larger than the others.

Humor recognition has been extensively studied with different methods in the past years. However, existing studies on humor recognition do not understand the mechanisms that generate humor. In this paper, inspired by the incongruity theory, any joke can be divided into two components (the setup and the punchline). Both components have multiple possible semantics, and there is an incongruous relationship between them. We use density matrices to represent the semantic uncertainty of the setup and the punchline, respectively, and design QE-Uncertainty and QE-Incongruity with the help of quantum entropy as features for humor recognition. The experimental results on the SemEval2021 Task 7 dataset show that the proposed features are more effective than the baselines for recognizing humorous and non-humorous texts.

Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors. The classical theory typically considers either random linear measurements or subsampled isometries and has found many applications, including accelerated magnetic resonance imaging, which is modeled by the subsampled Fourier transform. In this work, we develop a general theory of infinite-dimensional compressed sensing for abstract inverse problems, possibly ill-posed, involving an arbitrary forward operator. This is achieved by considering a generalized restricted isometry property, and a quasi-diagonalization property of the forward map. As a notable application, for the first time, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles $\theta_1,\dots,\theta_m$), which models computed tomography. In the case when the unknown signal is $s$-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove exact recovery under the condition \[ m\gtrsim s, \] up to logarithmic factors.

Gradient methods have become mainstream techniques for Bi-Level Optimization (BLO) in learning fields. The validity of existing works heavily rely on either a restrictive Lower- Level Strong Convexity (LLSC) condition or on solving a series of approximation subproblems with high accuracy or both. In this work, by averaging the upper and lower level objectives, we propose a single loop Bi-level Averaged Method of Multipliers (sl-BAMM) for BLO that is simple yet efficient for large-scale BLO and gets rid of the limited LLSC restriction. We further provide non-asymptotic convergence analysis of sl-BAMM towards KKT stationary points, and the comparative advantage of our analysis lies in the absence of strong gradient boundedness assumption, which is always required by others. Thus our theory safely captures a wider variety of applications in deep learning, especially where the upper-level objective is quadratic w.r.t. the lower-level variable. Experimental results demonstrate the superiority of our method.

We present an efficient quantum algorithm to simulate nonlinear differential equations with polynomial vector fields of arbitrary degree on quantum platforms. Models of physical systems that are governed by ordinary differential equations (ODEs) or partial differential equation (PDEs) can be challenging to solve on classical computers due to high dimensionality, stiffness, nonlinearities, and sensitive dependence to initial conditions. For sparse $n$-dimensional linear ODEs, quantum algorithms have been developed which can produce a quantum state proportional to the solution in poly(log(nx)) time using the quantum linear systems algorithm (QLSA). Recently, this framework was extended to systems of nonlinear ODEs with quadratic polynomial vector fields by applying Carleman linearization that enables the embedding of the quadratic system into an approximate linear form. A detailed complexity analysis was conducted which showed significant computational advantage under certain conditions. We present an extension of this algorithm to deal with systems of nonlinear ODEs with $k$-th degree polynomial vector fields for arbitrary (finite) values of $k$. The steps involve: 1) mapping the $k$-th degree polynomial ODE to a higher dimensional quadratic polynomial ODE; 2) applying Carleman linearization to transform the quadratic ODE to an infinite-dimensional system of linear ODEs; 3) truncating and discretizing the linear ODE and solving using the forward Euler method and QLSA. Alternatively, one could apply Carleman linearization directly to the $k$-th degree polynomial ODE, resulting in a system of infinite-dimensional linear ODEs, and then apply step 3. This solution route can be computationally more efficient. We present detailed complexity analysis of the proposed algorithms, prove polynomial scaling of runtime on $k$ and demonstrate the framework on an example.

Generalized correlation analysis (GCA) is concerned with uncovering linear relationships across multiple datasets. It generalizes canonical correlation analysis that is designed for two datasets. We study sparse GCA when there are potentially multiple generalized correlation tuples in data and the loading matrix has a small number of nonzero rows. It includes sparse CCA and sparse PCA of correlation matrices as special cases. We first formulate sparse GCA as generalized eigenvalue problems at both population and sample levels via a careful choice of normalization constraints. Based on a Lagrangian form of the sample optimization problem, we propose a thresholded gradient descent algorithm for estimating GCA loading vectors and matrices in high dimensions. We derive tight estimation error bounds for estimators generated by the algorithm with proper initialization. We also demonstrate the prowess of the algorithm on a number of synthetic datasets.

Many problems in computer science reduce to the recovery of an $n$-sparse measure from its (generalized) moments. Sparse measure recovery has been the research focus in super-resolution, tensor decomposition, and learning neural networks. The existing methods use either convex relaxations or overparameterization for recovery. Here, we propose recovery with non-convex optimization without overparameterization. Our algorithm is a (sub)gradient descent method optimizing a non-convex energy function studied in physics. We establish the global convergence of gradient descent on the energy function. This result enables us to solve super-resolution in $O(n^2)$ time, which significantly improves upon $O(n^3)$ time for solving convex relaxations. For a particular neural network, we prove the global convergence of subgradient descent on the population loss without overparameterization. The studied network has zero-one activations, and inputs drawn from the unit sphere.

During the inversion of discrete linear systems noise in data can be amplified and result in meaningless solutions. To combat this effect, characteristics of solutions that are considered desirable are mathematically implemented during inversion, which is a process called regularization. The influence of provided prior information is controlled by non-negative regularization parameter(s). There are a number of methods used to select appropriate regularization parameters, as well as a number of methods used for inversion. New methods of unbiased risk estimation and generalized cross validation are derived for finding spectral windowing regularization parameters. These estimators are extended for finding the regularization parameters when multiple data sets with common system matrices are available. It is demonstrated that spectral windowing regularization parameters can be learned from these new estimators applied for multiple data and with multiple windows. The results demonstrate that these modified methods, which do not require the use of true data for learning regularization parameters, are effective and efficient, and perform comparably to a learning method based on estimating the parameters using true data. The theoretical developments are validated for the case of two dimensional image deblurring. The results verify that the obtained estimates of spectral windowing regularization parameters can be used effectively on validation data sets that are separate from the training data, and do not require known data.

We propose a class of randomized quantum algorithms for the task of sampling from matrix functions, without the use of quantum block encodings or any other coherent oracle access to the matrix elements. As such, our use of qubits is purely algorithmic, and no additional qubits are required for quantum data structures. For $N\times N$ Hermitian matrices, the space cost is $\log(N)+1$ qubits and depending on the structure of the matrices, the gate complexity can be comparable to state-of-the-art methods that use quantum data structures of up to size $O(N^2)$, when considering equivalent end-to-end problems. Within our framework, we present a quantum linear system solver that allows one to sample properties of the solution vector, as well as an algorithm for sampling properties of ground states of Hamiltonians. As a concrete application, we combine these two sub-routines to present a scheme for calculating Green's functions of quantum many-body systems.

Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest A in a larger unitary transformation U that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of A, which is difficult in general, and not trivial even for well-structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well-structured sparse matrices, and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.