Hadi Salmasian

The Spectral Method for General Mixture Models

We present an algorithm for learning a mixture of distributions based on spectral projection. We prove a general property of spectral projection for arbitrary mixtures and show that the resulting algorithm is efficient when the components of the mixture are logconcave distributions in R n whose means are separated. The separation required grows with k, the number of components, and logn. This is the first result demonstrating the benefit of spectral projection for general Gaussians and widens the scope of this method. It improves substantially on previous results, which focus either on the special case of spherical Gaussians or require a separation that has a considerably larger dependence on n.

Unitary Representations of Nilpotent Super Lie Groups

We show that irreducible unitary representations of nilpotent super Lie groups can be obtained by induction from a distinguished class of sub super Lie groups. These sub super Lie groups are natural analogues of polarizing subgroups that appear in classical Kirillov theory. We obtain a concrete geometric parametrization of irreducible unitary representations by nonnegative definite coadjoint orbits. As an application, we prove an analytic generalization of the Stone-von Neumann theorem for Heisenberg-Clifford super Lie groups.

A notion of rank for unitary representations of reductive groups based on Kirillov's orbit method

We introduce a new notion of rank for unitary representations of semisimple groups over a local field of characteristic zero. The theory is based on Kirillov’s method of orbits for nilpotent groups over local fields. When the semisimple group is a classical group, we prove that the new theory is essentially equivalent to Howe’s theory of N-rank (see [Ho4], [L2], [Sc]). Therefore our results provide a systematic generalization of the notion of a small representation (in the sense of Howe) to exceptional groups. However, unlike previous works that used ad hoc methods to study different types of classical groups (and some exceptional ones; see [We], [LS]), our definition is simultaneously applicable to both classical and exceptional groups. The most important result of this article is a general “purity” result for unitary representations which demonstrates how similar partial results in these authors’ works should be formulated and proved for an arbitrary semisimple group in the language of Kirillov’s theory. The purity result is a crucial step toward studying small representations of exceptional groups. New results concerning small unitary representations of exceptional groups will be published in a forthcoming paper [S].

Smoothing operators and C∗-algebras for infinite dimensional Lie groups

A host algebra of a (possibly infinite dimensional) Lie group

Lie Supergroups, Unitary Representations, and Invariant Cones

G

Crossed product algebras and direct integral decomposition for Lie supergroups

is a

On an invariance property of the space of smooth vectors

C^*

Minimal dimension of faithful representations for p-groups

-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations

Small principal series and exceptional duality for two simply laced exceptional groups

\pi \colon G \to \U(\cH)

The spectral method for general mixture models

. In this paper we present a new approach to host algebras for infinite dimensional Lie groups which is based on smoothing operators, i.e., operators whose range is contained in the space