23 research outputs found
Towards a better approximation for sparsest cut?
We give a new -approximation for sparsest cut problem on graphs
where small sets expand significantly more than the sparsest cut (sets of size
expand by a factor bigger, for some small ; this
condition holds for many natural graph families). We give two different
algorithms. One involves Guruswami-Sinop rounding on the level- Lasserre
relaxation. The other is combinatorial and involves a new notion called {\em
Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which
we show exists in the input graph. Both algorithms run in time . We also show similar approximation algorithms in graphs with
genus with an analogous local expansion condition. This is the first
algorithm we know of that achieves -approximation on such general
family of graphs
On non-linear network embedding methods
As a linear method, spectral clustering is the only network embedding algorithm that offers both a provably fast computation and an advanced theoretical understanding. The accuracy of spectral clustering depends on the Cheeger ratio defined as the ratio between the graph conductance and the 2nd smallest eigenvalue of its normalizedLaplacian. In several graph families whose Cheeger ratio reaches its upper bound of Theta(n), the approximation power of spectral clustering is proven to perform poorly. Moreover, recent non-linear network embedding methods have surpassed spectral clustering by state-of-the-art performance with little to no theoretical understanding to back them.
The dissertation includes work that: (1) extends the theory of spectral clustering in order to address its weakness and provide ground for a theoretical understanding of existing non-linear network embedding methods.; (2) provides non-linear extensions of spectral clustering with theoretical guarantees, e.g., via different spectral modification algorithms; (3) demonstrates the potentials of this approach on different types and sizes of graphs from industrial applications; and (4)makes a theory-informed use of artificial networks
Pythagorean powers of hypercubes
For consider the -dimensional hypercube as equal to the
vector space , where is the field of size two.
Endow with the Hamming metric, i.e., with the metric induced
by the norm when one identifies with
. Denote by the
-fold Pythagorean product of , i.e., the space of all
, equipped with the metric
It is shown here that the
bi-Lipschitz distortion of any embedding of into
is at least a constant multiple of . This is achieved through
the following new bi-Lipschitz invariant, which is a metric version of (a
slight variant of) a linear inequality of Kwapie{\'n} and Sch\"utt (1989).
Letting denote the standard basis of the
space of all by matrices , say that a metric space
is a KS space if there exists such that for every , every mapping satisfies
\begin{equation*}\label{eq:metric KS abstract}
\frac{1}{n}\sum_{j=1}^n\mathbb{E}\left[d_X\Big(f\Big(x+\sum_{k=1}^ne_{jk}\Big),f(x)\Big)\right]\le
C \mathbb{E}\left[d_X\Big(f\Big(x+\sum_{j=1}^ne_{jk_j}\Big),f(x)\Big)\right],
\end{equation*} where the expectations above are with respect to and chosen
uniformly at random. It is shown here that is a KS space (with , which is best possible), implying the above nonembeddability
statement. Links to the Ribe program are discussed, as well as related open
problems.Comment: added section
Recommended from our members
Essays in transportation inequalities, entropic gradient flows and mean field approximations
This thesis consists of four chapters. In Chapter 1, we focus on a class of transportation inequalities known as the transportation-information inequalities. These inequalities bound optimal transportation costs in terms of relative Fisher information, and are known to characterize certain concentration properties of Markov processes around their invariant measures. We provide a characterization of the quadratic transportation-information inequality in terms of a dimension-free concentration property for i.i.d. copies of the underlying Markov process, identifying the precise high-dimensional concentration property encoded by this inequality. We also illustrate how this result is an instance of a general convex-analytic tensorization principle.
In Chapter 2, we study the entropic gradient flow property of McKean--Vlasov diffusions via a stochastic analysis approach. We formulate a trajectorial version of the relative entropy dissipation identity for these interacting diffusions, which describes the rate of relative entropy dissipation along every path of the diffusive motion. As a first application, we obtain a new interpretation of the gradient flow structure for the granular media equation. Secondly, we show how the trajectorial approach leads to a new derivation of the HWBI inequality.
In Chapter 3, we further extend the trajectorial approach to a class of degenerate diffusion equations that includes the porous medium equation. These equations are posed on a bounded domain and are subject to no-flux boundary conditions, so that their corresponding probabilistic representations are stochastic differential equations with normal reflection on the boundary. Our stochastic analysis approach again leads to a new derivation of the Wasserstein gradient flow property for these nonlinear diffusions, as well as to a simple proof of the HWI inequality in the present context.
Finally, in Chapter 4, we turn our attention to mean field approximation -- a method widely used to study the behavior of large stochastic systems of interacting particles. We propose a new approach to deriving quantitative mean field approximations for any strongly log-concave probability measure. Our framework is inspired by the recent theory of nonlinear large deviations, for which we offer an efficient non-asymptotic perspective in log-concave settings based on functional inequalities. We discuss three implications, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems