Given a matrix of observed data, Principal Components Analysis (PCA) computes a small number of orthogonal directions that contain most of its variability. Provably accurate solutions for PCA have been in use for a long time. However, to the best of our knowledge, all existing theoretical guarantees for it assume that the data and the corrupting noise are mutually independent, or at least uncorrelated. This is valid in practice often, but not always. In this paper, we study the PCA problem in the setting where the data and noise can be correlated. Such noise is often also referred to as ``data-dependent noise. We obtain a correctness result for the standard eigenvalue decomposition (EVD) based solution to PCA under simple assumptions on the data-noise correlation. We also develop and analyze a generalization of EVD, cluster-EVD, that improves upon EVD in certain regimes.

The remarkable success of transformers in sequence modeling tasks, spanning various applications in natural language processing and computer vision, is attributed to the critical role of self-attention.

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SLR, but an explicit connection between the two had not been made.

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Additionally, the framework is further applied to PCA problem to demonstrate the superiority and effectiveness of the proposedRWL-ANmodel.

Usingalgebraic geometry,weestablish that UPCA is a well-defined algebraic problem in the sense that the only matrices of minimal rank that agree with the given data are row-permutations of the ground-truth matrix, arising as the unique solutions of a polynomial system of equations.

A prime example is single-cell RNA sequencing (scRNA-seq) data (Figure 1).

Infact, the SDPmaybeinexacteven fork =3 (see Section 8). Suppose program to(2) (3) (4) (5) nonemptyfeasibleset.