arXiv subjects

Vandy Tombs

Publications and source records attributed to Vandy Tombs.

3 recordsLinked to original sources

arXiv2026

Testing the mixture model hypothesis via spectral gap

In this paper, we study the problem of testing whether or not a given probability measure $\mu$ on $\mathbb{R}^{d}$ can be decomposed as a mixture of two probability measures whose second order statistics are significantly different. We call this the problem of testing the mixture model hypothesis. To tackle it, we introduce a new set of computable orthogonal invariants of $\mu$, namely, the eigenvalues of the 4th moment operator $T_{\mu}$ associated with the measure. We prove that the largest eigenvalue is always an outlier eigenvalue. Further, we show how the first and second largest eigenvalues of $T_{\mu}$ give nonasymptotic bounds for this problem and give a complete resolution of the asymptotic version of the problem under the $L^{8}$-$L^{2}$ equivalence assumption.

math.PR↗

arXiv2020

Efficiently updating a covariance matrix and its LDL decomposition

Equations are presented which efficiently update or downdate the covariance matrix of a large number of $m$-dimensional observations. Updates and downdates to the covariance matrix, as well as mixed updates/downdates, are shown to be rank-$k$ modifications, where $k$ is the number of new observations added plus the number of old observations removed. As a result, the update and downdate equations decrease the required number of multiplications for a modification to $\Theta((k+1)m^2)$ instead of $\Theta((n+k+1)m^2)$ or $\Theta((n-k+1)m^2)$, where $n$ is the number of initial observations. Having the rank-$k$ formulas for the updates also allows a number of other known identities to be applied, providing a way of applying updates and downdates directly to the inverse and decompositions of the covariance matrix. To illustrate, we provide an efficient algorithm for applying the rank-$k$ update to the LDL decomposition of a covariance matrix.

math.NA↗

arXiv2017

Transfinitely valued Euclidean domains have arbitrary indecomposable order type

We prove that for every indecomposable ordinal there exists a (transfinitely valued) Euclidean domain whose minimal Euclidean norm is of that order type. Conversely, any such norm must have indecomposable type, and so we completely characterize the norm complexity of Euclidean domains. Modifying this construction, we also find a finitely valued Euclidean domain with no multiplicative integer valued norm.

math.AC↗