Algebraic Identifiability of Gaussian Mixtures

Abstract

We prove that all moment varieties of univariate Gaussian mixtures have the expected dimension. Our approach rests on intersection theory and Terracini's classification of defective surfaces. The analogous identifiability result is shown to be false for mixtures of Gaussians in dimension three and higher. Their moments up to third order define projective varieties that are defective. Our geometric study suggests an extension of the Alexander-Hirschowitz Theorem for Veronese varieties to the Gaussian setting. One-dimensional GaussiansThe moments m 0 , m 1 , m 2 , . . . , m d of a Gaussian distribution on the real line are polynomial expressions in the mean µ and the variance σ 2 . These expressions will be reviewed in Remark 5. They give a parametric representation of the Gaussian moment surface G 1,d in P d . The following implicit representation of that surface was derived in [2, Proposition 2].Proposition 3. Let d ≥ 3. The homogeneous prime ideal of the Gaussian moment surface G 1,d is minimally generated by d 3 cubics. These are the 3 × 3-minors of the 3 × d-matrixThe 3 × 3-minors of the matrix G d form a Gröbner basis for the prime ideal of G 1,d with respect to the reverse lexicographic term order. This implies that G 1,d has degree d 2 in P d . Our first new result concerns the singular locus on the Gaussian moment surface.Lemma 4. The singular locus of the surface G 1,d is the line defined by m 0 , m 1 , . . . , m d−2 .

Moment varieties of measures on polytopes

“…Ten of these irreducible modules in V are unique. The two irreducible GL 3 -modules with highest weight (2,3,4) are not unique.…”

“…For a concrete example, consider the quadrilateral X = {(1, −1), (3,2), (2,4), (−1, 2)}. The fiber for this X consists of 80 real points.…”

“…The ten unique GL 3modules get merged into two clusters, as seen in Table 2. Moreover, there is a unique way of choosing two GL 3 -modules with highest weight (2,3,4) such that acting with the affine group on one of these modules stays within one of the two clusters in Table 2.…”

Tapas of Algebraic Statistics

“…Recent progress with this approach has been made by Améndola et al [2,3], with partial answers. For example, it was shown [3] that considering all the moments up to order 3 − 1 will yield generically a finite number of Gaussian mixture densities with the same matching moments.…”

“…For example, it was shown [3] that considering all the moments up to order 3 − 1 will yield generically a finite number of Gaussian mixture densities with the same matching moments. In other words, the polynomial moment system generalizing (1) will generically have a finite number of solutions for the 3 unknown parameters , , for 1 ≤ ≤ .…”