S. Sra
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Registered User
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Researcher in Optimization, Machine Learning, etc.
Hobbyist in: Matrix Analysis
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1d |
revised |
Applications of Hilbert’s metric added new reference |
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1d |
accepted | On a version of gradient descent |
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1d |
answered | How to maximize the determinant of a matrix of the form VDV^H |
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1d |
revised |
How to maximize the determinant of a matrix of the form VDV^H added some tags, fixed TeX and some grammar plus typos |
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1d |
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On a version of gradient descent I tested it on a simple numerical example; the sequence $r_t$ is not monotonic. Since you essentially brought up this point, I attributed it to you. |
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1d |
revised |
On a version of gradient descent fixed answer |
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1d |
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On a version of gradient descent Ok, I think I was too quick!! I'll check the details of the proof tmw and update my answer. |
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1d |
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On a version of gradient descent Sorry, don't have time to type it in right now; please see (2.1.7 and 2.1.8) in the book Introductory lectures on convex optimization by Yu. Nesterov; you'll have to adapt those lemmas to the case of $f$ restricted to the $i$-th coordinate. From that the desired conclusion will follow. (Also, since $f$ is convex, $f'$ is monotone, so the inequality that you've mentioned in the comment also holds. The index being largest in magnitude was used for an earlier inequality in the cited paper. |
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2d |
answered | On a version of gradient descent |
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May 13 |
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Reference needed for: Automatic generation of relevant mathematical exercises (similar to ones written by human) with the help of machine learning "automatic", "relevant", "similar to ...", "mathematical" --- quite a challenging list I'd say. Maybe you could begin by creating a database of questions (taken e.g., from Polya and Szego's book)---but have you already figured out other basic issues such as: how'd you represent the input data? Maybe some work in automatic theorem proving might suggest potential automation clues.... |
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May 13 |
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Convexity of polylogaritms sorry; I made a typo in my calculations! this convexity seems rather delicate because a slight change seems to make it concave...I'll think about it if I get time; |
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May 13 |
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Collatz stopping-time and Poisson distribution, and connection to other problems? Did you mean "threads" or was it really "threats"? In both cases, however, the question remains the same :-) |
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May 11 |
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M-matrix plus S-matrix is P-matrix? Sorry, i just mistyped the extra word "integral" there...it is of course positive definite over the reals too! |
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May 11 |
awarded | ● Good Question |
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May 10 |
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M-matrix plus S-matrix is P-matrix? @Santiago: A symmetric, nonsingular $M$-matrix is a Stieltjes matrix and a symmetric matrix over the integers is Stieltjes if and only if it is positive definite. So at least the integral counterexamples above will not extend to the case when we have symmetry (as the sum of two strictly (symmetric) positive definite matrices cannot lose rank). |
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May 10 |
awarded | ● Nice Answer |
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May 10 |
awarded | ● Nice Answer |
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May 10 |
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M-matrix plus S-matrix is P-matrix? (but replacing the 1s by 1/4 and 3/4, in the M and S parts, respectively does the trick) |
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May 10 |
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M-matrix plus S-matrix is P-matrix? Will, you forgot to add the diagonals! |
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May 9 |
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What happens to Virasoro at c=25? @José: wow, so random turned out to be not entirely random---though unfortunately I am quite far from understanding the content and background that you have provided. Thanks! |
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May 8 |
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Literature on Exponential of a Quadratic Form It is neither convex nor concave (ignore for now the extra constraint that $x^Tx=1$, because with that constraint, you are asking for convexity on the surface of a hypersphere, which can at best hold only very locally); to see why, simply generate a few random vectors and test what happens to $f$. |
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May 8 |
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On solution of a class of discrete-time Lyapunov equation Also note that if $\|\sum_i F_i \kron F_i\| < 1$, then starting from $X_0=I$, you can iterate $X_{k+1} = \sum_i F_iX_kF_i^T + C$, and converge to the unique semidefinite solution. If the operators don't satisfy this sufficient condition, then more thought is needed. |
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May 8 |
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On solution of a class of discrete-time Lyapunov equation Ah, so you are asking whether $\text{mat}(x)$ will be positive semidefinite? Well, if the original linear system has a unique solution that is guaranteed to be semidefinite, then by solving the vectorized linear system, we should end up recovering that. It remains to determine conditions under which the original system has a semidefinite solution. The above system seems to me to be a well-studied problem; hopefully, F. Poloni (on MO) takes a look and provides some more details, as I think he knows more about these things. |
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May 8 |
accepted | On solution of a class of discrete-time Lyapunov equation |
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May 8 |
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What happens to Virasoro at c=25? Are these objects related to bosonic string theory with its $25+1$ space-time dimension (sorry, random pattern matching content free comment) :-) |
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May 7 |
answered | On solution of a class of discrete-time Lyapunov equation |
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May 3 |
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Uniqueness of fixed points for rational transformations I kept everything strictly positive to not have to deal with infinities and potentially unbounded derivatives of $\log$---but with care, usually one can get away :-) |
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May 2 |
accepted | Uniqueness of fixed points for rational transformations |
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May 1 |
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Norms on tensor products to get some comparison inequalities, you might benefit from searching for Pisier's survey on Grothendieck inequalities... |
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Apr 30 |
answered | Uniqueness of fixed points for rational transformations |
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Apr 29 |
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Number of Distinct Sums of Integers this sounds like an arithmetic combinatorics problem.... |
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Apr 28 |
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how do you call a function that satisfies the metric axioms except for the coincidence axiom? One example would be: $d(x,y) := \delta(x,y)+\delta(x,a)+\delta(y,b)$, where $\delta(x,y)$ is a usual metric, and $a \neq b$ are two arbitrary points in space (to have an example slightly less trivial than $d(x,y) := \delta(x,y) + \epsilon$... |
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Apr 28 |
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how do you call a function that satisfies the metric axioms except for the coincidence axiom? Hi Marco: I'd call these distances "strictly positive definite" but that sounds like mixing up terms. |
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Apr 28 |
revised |
A spectral inequality for positive-definite matrices fixed stupid error |
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Apr 28 |
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A spectral inequality for positive-definite matrices Oops, sorry, you are right, it should be $nM^2-m^2$! When typing it up, I felt the $\sqrt{M}$ factor to be a bit odd. Let me update the answer! Thanks. |
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Apr 28 |
accepted | A spectral inequality for positive-definite matrices |
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Apr 28 |
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Toolbox for directed graphs wrong forum in my opinion; you might want to try stats.stackexchange |
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Apr 27 |
answered | A spectral inequality for positive-definite matrices |
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Apr 26 |
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Fast way of finding RSS of Multiple Linear Regression mathoverflow.net/howtoask |
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Apr 26 |
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Finding a sub-matrix from a fat matrix with the best condition number. seems to be a duplicate of: mathoverflow.net/questions/104803/… |
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Apr 26 |
answered | Finding a sub-matrix from a fat matrix with the best condition number. |
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Apr 25 |
answered | iteratively (approximately) solving a sum of exponentials |
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Apr 24 |
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Constraint optimization problem for any dimensionality $n>1$. @Mark: maybe, though here I think first it'll be good to see how these constraints can be combined to yield a simpler problem---the cubic constraint is the troublesome one...) |
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Apr 24 |
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Constraint optimization problem for any dimensionality $n>1$. yes, i know that it is not "the" :-) --- (the second largest eigenvalue occurs twice) |
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Apr 23 |
answered | Constraint optimization problem for any dimensionality $n>1$. |
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Apr 22 |
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Global maximum of non-concave function There are several possible maxima it seems. Consider, e.g., for $f_1$, that $x_1=x_6=1/2$ and all other $x$s are zero. Then, $f_1=1$; same value is also achieved by your choice of $x_i=1/6$. Or are you asking whether for an optimum we can always pick a uniform vector? This might follows from the cyclic symmetry of your function, or the other kinds of symmetry groups that these functions are encoding. |
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Apr 22 |
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Another identity involving sums of (alternating) binomial coefficients. @Todd: Oren Patashnik is the name; also, that's the guy who (co-)made BibTeX, so .... :-) |
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Apr 22 |
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What are the most important open problems in algebraic combinatorics? in any case, this should be made community wiki... |
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Apr 22 |
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The eigenvalues of the product of two matrices @Douglas: ok, I think in my comment the word "any" was meant in a colloquial style to mean "pretty much any trivial", rather than in a mathematically precise sense! |
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Apr 19 |
revised |
Ratio sum comparison on operators updated the reflect that an alleged solution was wrong. |
