Related papers: Unique Sharp Local Minimum in $\ell_1$-minimizatio…

Sparse clustering, which aims to find a proper partition of an extremely high-dimensional data set with redundant noise features, has been attracted more and more interests in recent years. The existing studies commonly solve the problem in…

We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network (parameterized as $x \mapsto W_N W_{N-1} \cdots W_1 x$) by minimizing the $\ell_2$ loss over whitened data. Convergence at a linear…

Dictionary learning aims at seeking a dictionary under which the training data can be sparsely represented. Methods in the literature typically formulate the dictionary learning problem as an optimization w.r.t. two variables, i.e.,…

In over two decades of research, the field of dictionary learning has gathered a large collection of successful applications, and theoretical guarantees for model recovery are known only whenever optimization is carried out in the same…

We study the problem of learning a single neuron with respect to the $L_2^2$-loss in the presence of adversarial label noise. We give an efficient algorithm that, for a broad family of activations including ReLUs, approximates the optimal…

The power of sparse signal modeling with learned over-complete dictionaries has been demonstrated in a variety of applications and fields, from signal processing to statistical inference and machine learning. However, the statistical…

We propose an efficient online dictionary learning algorithm for kernel-based sparse representations. In this framework, input signals are nonlinearly mapped to a high-dimensional feature space and represented sparsely using a virtual…

Learning optimal dictionaries for sparse coding has exposed characteristic sparse features of many natural signals. However, universal guarantees of the stability of such features in the presence of noise are lacking. Here, we provide very…

We consider the minimization of the number of non-zero coefficients (the $\ell_0$ "norm") of the representation of a data set in terms of a dictionary under a fidelity constraint. (Both the dictionary and the norm defining the constraint…

Correlation Clustering is a powerful graph partitioning model that aims to cluster items based on the notion of similarity between items. An instance of the Correlation Clustering problem consists of a graph $G$ (not necessarily complete)…

Locally decodable codes (LDCs) are error-correcting codes $C : \Sigma^k \to \Sigma^n$ that admit a local decoding algorithm that recovers each individual bit of the message by querying only a few bits from a noisy codeword. An important…

We present preconditioned stochastic gradient descent (SGD) algorithms for the $\ell_1$ minimization problem $\min_{x}\|A x - b\|_1$ in the overdetermined case, where there are far more constraints than variables. Specifically, we have $A…

We investigate compressed sensing (CS) techniques for reducing the number of measurements in photoacoustic tomography (PAT). High resolution imaging from CS data requires particular image reconstruction algorithms. The most established…

In this paper, we carry out a unified study for $L_1$ over $L_2$ sparsity promoting models, which are widely used in the regime of coherent dictionaries for recovering sparse nonnegative/arbitrary signals. First, we provide a unified…

This paper considers correlation clustering on unweighted complete graphs. We give a combinatorial algorithm that returns a single clustering solution that is simultaneously $O(1)$-approximate for all $\ell_p$-norms of the disagreement…

Compressed sensing shows that a sparse signal can stably be recovered from incomplete linear measurements. But, in practical applications, some signals have additional structure, where the nonzero elements arise in some blocks. We call such…

We present Epidemic Learning (EL), a simple yet powerful decentralized learning (DL) algorithm that leverages changing communication topologies to achieve faster model convergence compared to conventional DL approaches. At each round of EL,…

This paper considers solving the unconstrained $\ell_q$-norm ($0\leq q<1$) regularized least squares ($\ell_q$-LS) problem for recovering sparse signals in compressive sensing. We propose two highly efficient first-order algorithms via…

Affine phase retrieval is the problem of recovering signals from the magnitude-only measurements with a priori information. In this paper, we use the $\ell_1$ minimization to exploit the sparsity of signals for affine phase retrieval,…

We study computationally and statistically efficient Reinforcement Learning algorithms for the linear Bellman Complete setting. This setting uses linear function approximation to capture value functions and unifies existing models like…