Papers/preprints:
- 1. Norms of random matrices: local and global problems (with R.Vershynin)
Submitted. Read on arxiv.org.
Can the behavior of a random matrix be improved by modifying a small fraction
of its entries? Consider a random matrix A with i.i.d. entries. We show that
the operator norm of A can be reduced to the optimal order \( O(n^{-1/2}) \) by zeroing
out a small submatrix of $A$ if and only if the entries have zero mean and finite
variance. Moreover, we obtain an almost optimal dependence between the size
of the removed submatrix and the resulting operator norm. Our approach utilizes
the cut norm and Grothendieck-Pietsch factorization for matrices, and it combines
the methods developed recently by C. Le and R. Vershynin and by E. Rebrova and
K. Tikhomirov.
- 2. Refined epsilon-nets and invertibility of random square matrices with i.i.d. heavy-tailed entries (with K.Tikhomirov)
Israel J. Math., to appear. Read on arxiv.org.
We consider an n × n random matrix with i.i.d. entries with zero mean and unit variance.
We show that for some L>0 and u∈[0, 1) the smallest singular value cannot be too small:
Prob(s_n(A) ≤ εn^{−1/2}) ≤ Lε + u^n for all ε > 0.
This result generalizes a theorem of Rudelson and Vershynin
that proves the same estimate for the square random matrices with subgaussian entries.
Undergraduate work:
For my masters thesis in Moscow State University, I was thinking about functions of bounded variation on
infinite dimensional spaces. I considered several classes of functions (bounded variation and bounded
semivariation), investigated their properties and compared them. My related publication is:
- Functions of bounded variation on infinite-dimensional spaces with measures (with V.Bogachev)
- Doklady Mathematics (Doklady Akademii Nauk), 87(2), 03/2013.