Gaussian Mixture Identifiability from degree 6 Moments

We resolve most cases of identifiability from sixth-order moments for Gaussian mixtures on spaces of large dimensions. Our results imply that the parameters of a generic mixture of $m\leq\mathcal{O}(n^4)$ Gaussians on $\mathbb R^n$ can be uniquely recovered from the mixture moments of degree 6. The constant hidden in the $\mathcal{O}$-notation is optimal and equals the one in the upper bound from counting parameters. We give an argument that degree-4 moments never suffice in any nontrivial case, and we conduct some numerical experiments indicating that degree 5 is minimal for identifiability.Comment: 22 page

Third Powers of Quadratics are generically Identifiable up to quadratic Rank

We consider the inverse problem for the map $$(S^2 (\mathbb C^n))^{m} \to S^6(\mathbb C^n), (q_{1},\ldots, q_{m}) \mapsto \sum_{i=1}^m q_{i}^3, \qquad (n, m \in \mathbb N)$$ which captures the moment problem for mixtures of centered Gaussians in the smallest interesting degree. We show that for any $n\in \mathbb N$, this map is generically one-to-one (up to permutations of $q_1,\ldots, q_m$) as long as $m\le {n\choose 2} + 1$, thus proving generic identifiability for mixtures of centered Gaussians from their (exact) moments of degree at most $6$ up to rank ${n\choose 2} + 1$. We rely on the study of tangent spaces of secant varieties and the contact locus.Comment: 14 pages. Code for the base case computations can be found on GitHu

Unique powers-of-forms decompositions from simple Gram spectrahedra

We consider simultaneous Waring decompositions: Given forms of degrees k*d, (d=2,3), which admit a representation as d-th power sums of k-forms, when is it possible to reconstruct the individual terms from the power sums? Such powers-of-forms decompositions model the moment problem for mixtures of centered Gaussians. The novel approach of this paper is to use semidefinite programming in order to perform a reduction to tensor decomposition. The proposed method works on typical parameter sets at least as long as m≤n−1, where m is the rank of the decomposition and n is the number of variables. While provably not tight, this analysis still gives the currently best known rank threshold for decomposing third order powers-of-forms, improving on previous work in both asymptotics and constant factors. Our algorithm can produce proofs of uniqueness for specific decompositions. A numerical study is conducted on Gaussian random trace-free quadratics, giving evidence that the success probability converges to 1 in an average case setting, as long as m=n and n→∞. Some evidence is given that the algorithm also succeeds on instances of rank quadratic in the dimension

Gaussian mixture identifiability from degree 6 moments

We resolve most cases of identifiability from sixth-order moments for Gaussian mixtures on spaces of large dimensions. Our results imply that the parameters of a generic mixture of m~Θ(n^4) Gaussians on ℝ^n can be uniquely recovered from the mixture moments of degree 6. The constant hidden in the O-notation is optimal and equals the one in the upper bound from counting parameters. We give an argument that degree-4 moments never suffice in any nontrivial case, and we conduct some numerical experiments indicating that degree 5 is minimal for identifiability

Unique powers-of-forms decompositions from simple Gram spectrahedra

We consider simultaneous Waring decompositions: Given forms $f_d$ of degrees $kd$, $(d = 2,3 )$, which admit a representation as $d$-th power sums of $k$-forms $q_1,\ldots,q_m$, when is it possible to reconstruct the addends $q_1,\ldots,q_m$ from the power sums $f_d$? Such powers-of-forms decompositions model the moment problem for mixtures of centered Gaussians. The novel approach of this paper is to use semidefinite programming in order to perform a reduction to tensor decomposition. The proposed method works on typical parameter sets at least as long as $m\leq n-1$, where $m$ is the rank of the decomposition and $n$ is the number of variables. While provably not tight, this analysis still gives the currently best known rank threshold for decomposing third order powers-of-forms, improving on previous work in both asymptotics and constant factors. Our algorithm can produce proofs of uniqueness for specific decompositions. A numerical study is conducted on Gaussian random trace-free quadratics, giving evidence that the success probability converges to $1$ in an average case setting, as long as $m = n$ and $n\to \infty$. Some evidence is given that the algorithm also succeeds on instances of rank $m = \Theta(n^2)$.Comment: 25 pages, 2 figures. Accompanying code may be found on GitHu

Gaussian Mixture Identifiability from degree-6 moments

This repository accompanies the arxiv submission Gaussian Mixture Identifiability from degree-6 moments. It contains code and data from numerical experiments, and parts of proofs that were verified on a computer, as well as the code that was used to generate the plots

Code and files for: Unique powers-of-forms decompositions from simple Gram spectrahedra

This repository accompanies the paper Unique powers-of-forms decompositions from simple Gram spectrahedra. It contains: a tentative implementation of Algorithm 1 therein for powers-of-forms decomposition, including subroutines to: verify whether a form is uniquely Sum-of-Squares representable. compute the Sum-of-Squares support of a given SOS form. decompose positively weighted power sums of linear forms. data from the numerical experiments of Section 4.2