We'll discuss a new generalization of the Cheeger inequality to higher-order structures in networks including network motifs. This is easy to implement and seemlessly generalizes spectral clustering to directed, signed, and many other types of complex networks. In particular, our generalization allow us to re-use the large history of existing ideas in spectral clustering including local methods, overlapping methods, and relationships with kernel k-means.

Secure multi-party computation enables mutually distrusting parties to compute over their private data. Such secure computation protocols leverage the security of atomic "building blocks" to perform complex computations privately. The security of these protocols crucially hinges on the assumption that these underlying building blocks have perfect security. Even minor imperfections in these building blocks can violate the security of these protocols.
This raises the following important question: "Can secure computation be based on imperfect building blocks?"
In this talk, I will present algorithmic solutions that resolve this question in the affirmative

We present elementary method for obtaining the spectral decomposition of hypermatrices generated by arbitrary combinations of Kronecker products and direct sums of cubic hypermatrices having side length 2. The method is based on a generalization of Parseval's identity. We use the general formulation of Parseval's identity to introduce hypermatrix Fourier transforms and discrete Fourier hypermatrices. We extend to hypermatrices orthogonalization procedures and Sylvester's classical Hadamard matrix construction. We conclude the talk with illustrations of spectral decompositions of adjacency hypermatrices of finite groups and a proof of a hypermatrix Rayleigh quotient inequality. This is a joint work with Yuval Filmus.