14,700 reputation
32979
bio website suvrit.de
location Internet
age
visits member for 4 years, 9 months
seen 2 hours ago

Researcher in Optimization and Machine Learning.

Hobbyist in Inequalities, Matrix Analysis, Combinatorics, Algebra, etc.

I'm here mostly to learn mathematics!


3h
comment The sum of squared logarithms conjecture
Oh I did not realize that the paper I saw two years ago was by you and your co-authors! I hope MO helps prove it.
8h
comment The sum of squared logarithms conjecture
This conjecture is a really nice one; I saw it a couple of years ago, but did not yet get time to think about it! In particular, it states that if eigenvalues of a positive definite matrix $X$ are $E$-majorized (to be defined) by those of $Y$, then $d(X,I) \le d(Y,I)$, where $d(\cdot,\cdot)$ is the Riemannian distance on the manifold of positive definite matrices (E-majorization is just the majorization in terms of elementary symmetric functions as noted in the conjecture above)
8h
comment Hadamard Product and Eigendecomposition
@mikitov: if $B$ is special, then surely one can say more; e.g., if $B=11^T$ :-) One can say more precise things about rougher quantities such as $\|A\circ B\|$ etc., or singular value inequalities, and so on; but a fine-grained description of the spectrum seems too hard.
11h
answered Hadamard Product and Eigendecomposition
11h
comment Hadamard Product and Eigendecomposition
the obvious answer is: no there won't be a nontrivial closed form.
22h
comment An inequality for copulas
Quite an interesting question. Once I get some time, I'll think about it again. I'm hoping that the Laplace transform representation helps achieve the proof...
2d
comment An inequality for copulas
My answer establishes the claim for $t \ge 1$. So I guess the question is now for general CM $f$...
May
25
comment Concavity of mixed volumes and mixed discriminants
Ok, that clarifies my doubts. Before plowing into this, however, I'd like to ask you do you have any numerical evidence that suggests that your conjecture may hold? Thanks!
May
25
comment Solving nonlinear inequality that involves norm2 operator
It is not clear what is it that you are trying to solve. What you have written is an inequality, not an equation. In particular, since $C, q, Z, p$ are all fixed, I don't see why making $\psi \to 0$ will not solve the inequality. I must be missing something...
May
25
comment Concavity of mixed volumes and mixed discriminants
The closest paper I found relevant to your question is: arxiv.org/pdf/1412.8200.pdf --- does Corollary 4.1 in there help?
May
25
comment Concavity of mixed volumes and mixed discriminants
The discussion on pg. 391 of ams.org/journals/bull/2002-39-03/S0273-0979-02-00941-2/… seems to contain useful pointers; the conjecture seems nice but probably it breaks.
May
25
comment John Nash's Mathematical Legacy
I hope somebody with more expertise chimes up and summarizes Nash's work on cooperative games (I had the fortune to attend a talk by him in 2008 where he carefully described his cooperative games model, he almost expressed a personal relation with each of the variables on his slides)...
May
24
comment Solving nonlinear inequality that involves norm2 operator
the question is not clear; what about $\psi=0$?
May
20
comment complexity of eigenvalue decomposition
@SébastienLoisel: I think you are right to worry about the true computational complexity; I am not sure if the paper I linked to manages to avoid intermediate coefficient growth --- so your skepticism is justified...
May
20
comment complexity of eigenvalue decomposition
@IgorRivin: no, as also Robert notes, the unitary eigenproblem is as easy/hard as the general case (so it can be done in $O(n^3)$.
May
19
comment complexity of eigenvalue decomposition
Please have a look at the paper: "The complexity of the matrix eigenproblem" by Victor Pan and Zhao Chen. Their Theorem 1 contradicts your claims...
May
15
comment Sum identities with immanants
That result is an adaptation of en.wikipedia.org/wiki/Schur_orthogonality_relations -- so that in particular, if $G$ is a finite group and $\chi, \xi$ are irreducible chars, then $$\sum_{\sigma \in G} \overline{\chi(\sigma)}\xi(\sigma\tau) = \begin{cases} |G|\chi(\tau)/\chi(e), & \text{if}\ \chi=\xi;\\ 0 & \text{otherwise}.\end{cases}$$
May
15
answered Sum identities with immanants
May
13
comment Convexity of the product of two exponential matrices
Although the original claim is false (Mike Jury has an excellent explicit counterexample), a much weaker related claim is true. Claim. Let $A \in M_n(\mathbb{C})$. The function $f: [0,c] \to \mathbb{R}_+$ defined by \begin{equation*} f(t) = \|e^{tA}\| \end{equation*} is convex for any unitarily invariant norm $\|\cdot\|$.
May
12
comment $\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?
More generally, also directly relevant are pp. 419--421 of Watson's "Treatise on Bessel functions"