Low rank approximation of matrices has been well studied in literature. Singular value decomposition , QR decomposition with column pivoting, rank revealing QR factorization (RRQR), Interpolative decomposition etc are classical deterministic algorithms for low rank approximation. But these techniques are very expensive (O(n 3) operations are required for n × n matrices). There are several rando...

We present new methods for determining polynomials in the ideal of the variety of bilinear maps of border rank at most r. We apply these methods to several cases including the case r = 6 in the space of bilinear maps C × C → C. This space of bilinear maps includes the matrix multiplication operator M2 for two by two matrices. We show these newly obtained polynomials do not vanish on the matrix ...

We present a new analysis of sparse oblivious subspace embeddings, based on the ”matrix Chernoff” technique. These are probability distributions over (relatively) sparse matrices such that for any d-dimensional subspace of R, the norms of all vectors in the subspace are simultaneously approximately preserved by the embedding with high probability–typically with parameters depending on d but not...

Abstract. It is known that a high order tensor does not necessarily have an optimal low rank approximation, and that a tensor might not be orthogonally decomposable (i.e., admit a tensor SVD). We provide several sufficient conditions which lead to the failure of the tensor SVD, and characterize the existence of the tensor SVD with respect to the Higher Order SVD (HOSVD) of a tensor. In face of ...

In this paper we answer the question posed by M. Atiyah, see [12], and give an explicit formula for Floer homology of Brieskorn homology spheres in terms of their branching sets over the 3–sphere. We further show how Floer homology is related to other invariants of knots and 3–manifolds, among which are the μ̄–invariant of W. Neumann and L. Siebenmann and the Jones polynomial. Essential progress...

In this paper, we provide new yield-aware mask strategies to mitigate emerging variability and defectivity challenges. To address variability, we analyze CD variability with respect to reticle size, and its impact on parametric yield. With a cost model that incorporates mask, wafer, and processing cost considering throughput, yield, and manufacturing volume, we assess various reticle strategies...

Differential privacy is a promising privacy-preserving paradigm for statistical query processing over sensitive data. It works by injecting random noise into each query result, such that it is provably hard for the adversary to infer the presence or absence of any individual record from the published noisy results. The main objective in differentially private query processing is to maximize the...

We show that for each n and m, there is an existential first order sentence which is NOT logically equivalent to a sentence of quantifier rank at most m in infinitary logic augmented with all generalized quantifiers of arity at most n. We use this to show the strictness of the quantifier rank hierarchies for various logics ranging from existential (or universal) fragments of first order logic t...

Principal component analysis (PCA) requires the computation of a low-rank approximation to a matrix containing the data being analyzed. In many applications of PCA, the best possible accuracy of any rank-deficient approximation is at most a few digits (measured in the spectral norm, relative to the spectral norm of the matrix being approximated). In such circumstances, existing efficient algori...

Having shown its good performance in many applications, variational Bayesian (VB) learning is known to be one of the best tractable approximations to Bayesian learning. However, its performance was not well understood theoretically. In this paper, we clarify the behavior of VB learning in probabilistic PCA (or fully-observed matrix factorization). More specifically, we establish a necessary and...