Alexander Kushkuley

Alexander Kushkuley

Let $ χ$ be a character of a complex irreducible representation of a finite group $G$. We present a simple formula for the expectation of the random variable $(|χ|/χ(1))^{t} $ in terms of character ratios $ (|χ(g)|/χ(1))^{t}, \; g \in G, \; t \geq 0 $. As a follow up we briefly discuss asymptotic properties of the formula and its relation to the growth of dimensions of isotypic components in (virtual) tensor powers of irreducible representations

Alexander Kushkuley, Joshua Correa

Most if not all on-line item-to-item recommendation systems rely on estimation of a distance like measure (rank) of similarity between items. For on-line recommendation systems, time sensitivity of this similarity measure is extremely important. We observe that recommendation "recency" can be straightforwardly and transparently maintained by iterative reduction of ranks of inactive items. The paper briefly summarizes algorithmic developments based on this self-explanatory observation. The basic idea behind this work is patented in a context of online recommendation systems.

Alexander Kushkuley

A very simple event frequency approximation algorithm that is sensitive to event timeliness is suggested. The algorithm iteratively updates categorical click-distribution, producing (path of) a random walk on a standard $n$-dimensional simplex. Under certain conditions, this random walk is self-similar and corresponds to a biased Bernoulli convolution. Algorithm evaluation naturally leads to estimation of moments of biased (finite and infinite) Bernoulli convolutions.

Alexander Kushkuley

Estimation of top singular values is one of the widely used techniques and one of the intensively researched problems in Numerical Linear Algebra and Data Science. We consider here two general questions related to this problem: How top singular values are affected by zeroing out a sparse rectangular block of a matrix? How much top singular values differ from top column norms of a tall sparse non-negative matrix ?