We consider the following oblivious sketching problem: given ∈ (0, 1/3) and n ≥ d/ 2, design a distribution D over Rk×nd and a function f : R × R → R, so that for any n× d matrix A, Pr S∼D [(1− )‖A‖op ≤ f(S(A), S) ≤ (1 + )‖A‖op] ≥ 2/3, where ‖A‖op = supx:‖x‖2=1 ‖Ax‖2 is the operator norm of A and S(A) denotes S ·A, interpreting A as a vector in R. We show a tight lower bound of k = Ω(d2/ 2) for...

We affirmatively settle the question on existence of a real-valued higher order spectral shift function for pair self-adjoint operators $H$ and $V$ such that is bounded $V(H-iI)^{-1}$ belongs to Schatten–von Neumann ideal $\mathcal{S}^n$ compact in separable Hilbert space. also show satisfies same trace formula as known case $V\in\mathcal{S}^n$ it unique up polynomial summand $n-1$. Our result ...

The related inequality (1.1) was obtained by Maher [3, Theorem 3.2] who showed that, if A is normal and AT = TA, where T ∈ Cp , then ‖T − (AX −XA)‖p ≥ ‖T‖p for all X ∈ ( ), where Cp is the von Neumann-Schatten class, 1≤ p <∞, and ‖·‖p its norm. Here we show that Maher’s result is also true in the case where Cp is replaced by ( ), the ideal of all compact operators with ‖·‖∞ its norm.Which allow...

We study non-linear traces of the Choquet type and Sugeno on algebra compact operators. They have certain partial additivities. show that these additivities characterize both respectively. There exists a close relation between majorization theory. trace class operators for type. More generally we discuss Schatten–von Neumann p-class determine when they form Banach spaces. This is an attempt at ...

We study a new class of structured Schatten norms for tensors that includes two recently proposed norms (“overlapped” and “latent”) for convex-optimization-based tensor decomposition. Based on the properties of the structured Schatten norms, we analyze the performance of “latent” approach for tensor decomposition, which was empirically found to perform better than the “overlapped” approach in s...

MICHAEL K. TIPPETTy, STEPHEN E. COHNz, RICARDO TODLINGx, AND DAN MARCHESIN{ Abstract. The Schatten p-norm condition of the discrete-time Lyapunov operator LA defined on matrices P 2 Rn n by LAP P APAT is studied for stable matrices A 2 Rn n . Bounds are obtained for the norm of LA and its inverse that depend on the spectrum, singular values and radius of stability of A. Since the solution P of ...