All Questions
10 questions
42
votes
3
answers
5k
views
The probability for a symmetric matrix to be positive definite
Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio $p_n=...
- 52.3k
25
votes
1
answer
4k
views
What kind of random matrices have rapidly decaying singular values?
I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the $...
- 118k
3
votes
1
answer
3k
views
Singular value decomposition of random rectangular matrices
Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance).
What is the ...
- 884
18
votes
1
answer
1k
views
How fast can extreme eigenvalues of the average of random matrices converge to their expectation?
Suppose that $X_1,X_2,\ldots,X_m$ are independent $d\times d$ random matrices and let $\overline{X} := \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices is ...
- 181
10
votes
2
answers
1k
views
Probability of random (0,1) Toeplitz matrix being invertible
A Toeplitz matrix or diagonal-constant matrix is a matrix in which each descending diagonal from left to right is constant.
What is the probability that a random $n \times n$ binary Toeplitz ...
user32786
8
votes
0
answers
232
views
Decay of orthogonal contributions in a random set of vectors
Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$:
$$\frac{v_1}{\|v_1\|},\...
- 2,252
5
votes
0
answers
235
views
Riemann theta function inequality for a class of large random matrices
The following is essentially the same question as in this previous post, but since I have completely re-formulated it (hopefully for the better ;-), I decided to post a new question instead of an edit....
- 686
3
votes
0
answers
202
views
Difficult Gaussian-sum inequality for large random Bernoulli-Toeplitz matrices
I have come across the following problem in an attempt to prove an entropy bound for large random Bernoulli-Toeplitz matrices (Conjecture 1 on p. 16 of this preprint by Clifford et al. 2015), which is ...
- 686
2
votes
1
answer
905
views
Diagonalizability of Gaussian random matrices
Let $X$ be an $n\times n$ matrix whose elements are i.i.d. sampled from a normal distribution of zero mean and unit variance. Is $X$ diagonalizable over $\mathbb{C}$ with probability 1? Is there a ...
- 123
1
vote
1
answer
84
views
Asymptotic property of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional data matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\top.
...
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