<p>This thesis is concerned with the problem of characterizing the orbits of certain probability density functions under the action of the Euclidean group. Our motivating application is the recognition of a point configuration where the coordinates of the points are measured under noisy conditions. Consider a random variable X in R<sup>d</sup> with probability density function ρ(x). Let x<sub>1</sub> and x<sub>2</sub> be independent random samples following ρ(x). Define ∆ as the squared Euclidean distance between x<sub>1</sub> and x<sub>2</sub>. It has previously been shown that two distributions ρ(x) and ρ(x) consisting of Dirac delta distributions in generic positions that have the same respective distributions of ∆ are necessarily related by a rigid motion. That is, there exists some rigid motion g in the Euclidean group E(d) such that ρ(x) = ρ(g · x) for all x ∈ R<sup>d</sup> . To account for noise in the measurements, we assume X is a random variable in R<sup>d</sup> whose density is a k-component mixture of Gaussian distributions with means in generic position. We further assume that the covariance matrices of the Gaussian components are equal and of the form Σ = σ<sup>2</sup>1<sub>d</sub> with 0 ≤ σ<sup>2</sup> ∈ R. In Theorem 3.1.1 and Theorem 3.2.1, we prove that, when σ<sup>2</sup> is known, generic k-component Gaussian mixtures are uniquely reconstructible up to a rigid motion from the density of ∆. A more general formulation is proven in Theorem 3.2.3. Similarly, when σ<sup>2</sup> is unknown, we prove in Theorem 4.1.1 and Theorem 4.1.2 that generic equally-weighted k-component Gaussian mixtures with k = 1 and k = 2 are uniquely reconstructible up to a rigid motion from the distribution of ∆. There are at most three non-equivalent equally weighted 3-component Gaussian mixtures up to a rigid motion having the same distribution of ∆, as proven in Theorem 4.1.3. In Theorem 4.1.4, we present a test to check if, for a given k and d, the number of non-equivalent equally-weighted k-component Gaussian mixtures in R<sup>d</sup> having the same distribution of ∆ is at most (k choose 2) + 1. Numerical computations showed that distributions with k = 4, 5, 6, 7 such that d ≤ k −2 and (k, d) = (8, 1) pass the test, and thus have a finite number of reconstructions up to a rigid motion. When σ<sup>2</sup> is unknown and the mixture weights are also unknown, we prove in Theorem 4.2.1 that there are at most four non-equivalent 2-component Gaussian mixtures up to a rigid motion having the same distribution of ∆. </p>
| Identifer | oai:union.ndltd.org:purdue.edu/oai:figshare.com:article/22684960 |
| Date | 25 April 2023 |
| Creators | Kindyl Lu Zhao King (15347239) |
| Source Sets | Purdue University |
| Detected Language | English |
| Type | Text, Thesis |
| Rights | CC BY 4.0 |
| Relation | https://figshare.com/articles/thesis/ON_RECONSTRUCTING_GAUSSIAN_MIXTURES_FROM_THE_DISTANCE_BETWEEN_TWO_SAMPLES_AN_ALGEBRAIC_PERSPECTIVE/22684960 |
Page generated in 0.003 seconds