We consider the problem of learning multi-ridge functions of the form f(x) = g(Ax) from point evaluations of f . We assume that the function f is defined on an `2-ball in R, g is twice continuously differentiable almost everywhere, and A ∈ Rk×d is a rank k matrix, where k d. We propose a randomized, polynomial-complexity sampling scheme for estimating such functions. Our theoretical development...

We present the first range result for the total K-theory of C∗-algebras. This invariant has been used successfully to classify certain separable, nuclear C∗-algebras of real rank zero. Our results complete the classification of the so-called AD algebras of real rank zero.

we give further results for perron-frobenius theory on the numericalrange of real matrices and some other results generalized from nonnegative matricesto real matrices. we indicate two techniques for establishing the main theorem ofperron and frobenius on the numerical range. in the rst method, we use acorresponding version of wielandt's lemma. the second technique involves graphtheory.

Denote by W (A) the numerical range of a bounded linear operator A, and [A, B] = AB −BA the Lie product of two operators A and B. Let H, K be complex Hilbert spaces of dimension ≥ 2 and Φ : B(H) → B(K) be a map whose range contains all operators of rank ≤ 1. It is shown that Φ satisfies that W ([Φ(A), Φ(B)]) = W ([A, B]) for any A, B ∈ B(H) if and only if dim H = dim K, there exist ε ∈ {1,−1}, ...

Bringmann and Dousse recently established a conjecture of Dyson dealing with the limiting asymptotics of the Andrews-Garvan crank statistic for integer partitions. A direct “sieving” technique is used to establish this conjecture and establish the range of validity. Unlike the approach of Bringmann and Dousse, the technique readily yields the analogous result for Dyson’s partition rank and all ...

Abstract. We derive and study a Gauss-Newton method for computing a symmetric low-rank product XXT, where X ∈ Rn×k for k < n, that is the closest to a given symmetric matrix A ∈ Rn×n in Frobenius norm. When A = BTB (or BBT), this problem essentially reduces to finding a truncated singular value decomposition of B. Our Gauss-Newton method, which has a particularly simple form, shares the same or...

Stampfli has shown that for a given T £ B(H) there exists a K £ C(H) so that o(T + K) = ow(T). An analogous result holds for the essential numerical range We(T). A compact operator K is said to preserve the Weyl spectrum and essential numerical range of an operator T £ B(H) if o(T + K) = o„(T) and W(T + K)= We(T). Theorem. For each block diagonal operator T, there exists a compact operator K wh...