Hierarchical Orthogonal Matrix Generation and Matrix-Vector Multiplications in Rigid Body Simulations

In this paper, we apply the hierarchical modeling technique and study some numerical linear algebra problems arising from the Brownian dynamics simulations of biomolecular systems where molecules are modeled as ensembles of rigid bodies. Given a rigid body $p$ consisting of $n$ beads, the $6 \times 3n$ transformation matrix $Z$ that maps the force on each bead to $p$'s translational and rotational forces (a $6\times 1$ vector), and $V$ the row space of $Z$, we show how to explicitly construct the $(3n-6) \times 3n$ matrix $\tilde{Q}$ consisting of $(3n-6)$ orthonormal basis vectors of $V^{\perp}$ (orthogonal complement of $V$) using only $\mathcal{O}(n \log n)$ operations and storage. For applications where only the matrix-vector multiplications $\tilde{Q}{\bf v}$ and $\tilde{Q}^T {\bf v}$ are needed, we introduce asymptotically optimal $\mathcal{O}(n)$ hierarchical algorithms without explicitly forming $\tilde{Q}$. Preliminary numerical results are presented to demonstrate the performance and accuracy of the numerical algorithms

Visualization of AE's Training on Credit Card Transactions with Persistent Homology

Auto-encoders are among the most popular neural network architecture for dimension reduction. They are composed of two parts: the encoder which maps the model distribution to a latent manifold and the decoder which maps the latent manifold to a reconstructed distribution. However, auto-encoders are known to provoke chaotically scattered data distribution in the latent manifold resulting in an incomplete reconstructed distribution. Current distance measures fail to detect this problem because they are not able to acknowledge the shape of the data manifolds, i.e. their topological features, and the scale at which the manifolds should be analyzed. We propose Persistent Homology for Wasserstein Auto-Encoders, called PHom-WAE, a new methodology to assess and measure the data distribution of a generative model. PHom-WAE minimizes the Wasserstein distance between the true distribution and the reconstructed distribution and uses persistent homology, the study of the topological features of a space at different spatial resolutions, to compare the nature of the latent manifold and the reconstructed distribution. Our experiments underline the potential of persistent homology for Wasserstein Auto-Encoders in comparison to Variational Auto-Encoders, another type of generative model. The experiments are conducted on a real-world data set particularly challenging for traditional distance measures and auto-encoders. PHom-WAE is the first methodology to propose a topological distance measure, the bottleneck distance, for Wasserstein Auto-Encoders used to compare decoded samples of high quality in the context of credit card transactions.Comment: arXiv admin note: substantial text overlap with arXiv:1905.0989

Faster Rates for the Frank-Wolfe Method over Strongly-Convex Sets

The Frank-Wolfe method (a.k.a. conditional gradient algorithm) for smooth optimization has regained much interest in recent years in the context of large scale optimization and machine learning. A key advantage of the method is that it avoids projections - the computational bottleneck in many applications - replacing it by a linear optimization step. Despite this advantage, the known convergence rates of the FW method fall behind standard first order methods for most settings of interest. It is an active line of research to derive faster linear optimization-based algorithms for various settings of convex optimization. In this paper we consider the special case of optimization over strongly convex sets, for which we prove that the vanila FW method converges at a rate of $\frac{1}{t^2}$. This gives a quadratic improvement in convergence rate compared to the general case, in which convergence is of the order $\frac{1}{t}$, and known to be tight. We show that various balls induced by $\ell_p$ norms, Schatten norms and group norms are strongly convex on one hand and on the other hand, linear optimization over these sets is straightforward and admits a closed-form solution. We further show how several previous fast-rate results for the FW method follow easily from our analysis