Search

This paper contains corrections to Madea, Rosenberg, Torres-Ardila, "The Geometry of Loop Spaces II: Characteristic Classes," Advances in Math. (287), 2016, 485-518. The main change is that results about $\pi_1({\rm Diff}(M))$ are replaced by results about $\pi_1({\rm Isom}(M))$, where Diff$(M)$, Isom$(M)$ refer to the diffeomorphism and isometry group of the manifold $M$.

In this note, we compute the reproducing kernel for the RKHS of functions on $\mathbb{R}^n$ in a sufficiently high Sobolev norm.

We give examples of spin $4$-manifolds with boundary that do not admit metrics of positive scalar curvature and nonnegative mean curvature. These manifolds in fact have the stronger property that the conformal Laplacian with appropriate boundary conditions is never positive. The obstruction to the positivity of the conformal Laplacian is given by a real-valued $\xi$-invariant associated to the APS theorem for the twisted Dirac operator. We use analytic techniques related to the prescribed scalar curvature problem in conformal geometry to move beyond earlier work where the metric is a product near the boundary.

While natural gradients have been widely studied from both theoretical and empirical perspectives, we argue that some fundamental theoretical issues regarding the existence of gradients in infinite dimensional function spaces remain underexplored. We address these issues by providing a geometric perspective and mathematical framework for studying natural gradient that is more complete and rigorous than existing studies. Our results also establish new connections between natural gradients and RKHS theory, and specifically to the Neural Tangent Kernel (NTK). Based on our theoretical framework, we derive a new family of natural gradients induced by Sobolev metrics and develop computational techniques for efficient approximation in practice. Preliminary experimental results reveal the potential of this new natural gradient variant.

We introduce an iterative scheme to solve the Yamabe equation $ - a\Delta_{g} u + S u = \lambda u^{p-1} $ on small domains $(\Omega,g)\subset {\mathbb R}^n$ equipped with a Riemannian metric $g$. Thus $g$ admits a conformal change to a constant scalar curvature metric. The proof does not use the traditional functional minimization. Applications to the Yamabe problem on closed manifolds, manifolds with boundary, and noncompact manifolds are given in forthcoming papers.

We study the diffeomorphism and isometry groups of manifolds $\overline {M_p}$, $p\in\mathbb Z$, which are circle bundles over a closed $4n$-dimensional integral symplectic manifold. Equivalently, $\overline{M_p}$ is a compact $(4n+1)$-dimensional contact manifold with closed Reeb orbits. We use Wodzicki-Chern-Simons forms to prove that $\pi_1({\rm Diff}(\overline{M_p})$ and $\pi_1({\rm Isom}(\overline{M_p}))$ are infinite for $|p| \gg 0.$ For the Kodaira-Thurston manifold, we explicitly compute that this result holds for all $p$. We also give the first examples of nonvanishing Wodzicki-Pontryagin forms.

Commercial buildings account for approximately 35% of total US electricity consumption, of which nearly two-thirds is met by fossil fuels resulting in an adverse impact on the environment. This adverse impact can be mitigated by lowering energy consumption via control of occupant plugload usage in a closed-loop building environment. In this work, we conducted multiple experiments to analyze changes in occupant plugload energy consumption due to incentives and/or visual feedback. The incentives entailed daily monetary values between $5 and $50 administered in a randomized order and the visual feedback consisted of a web-based dashboard aimed at increasing the energy awareness of participants. Experiments were performed in government office and university buildings at NASA Ames Research Park located in Moffett Field, CA. Autoregressive models were constructed to predict expected plugload savings in the presence of exogenous variables. Analysis of the data revealed modulation of plugload energy consumption can be achieved via visual feedback and incentive mechanisms suggesting that occupant-in-the-loop control architectures may be effective in the commercial building environment. Our findings indicate that the mean energy reduction due to visual feedback in office and university environments were ~9.52% and ~21.61%, respectively. By augmenting the visual feedback in the university environment with a monetary incentive, the mean energy reduction was found to be ~24.22%, Comment: 22 pages, 5 figures, 6 tables, accepted version

While the study of a single network is well-established, technological advances now allow for the collection of multiple networks with relative ease. Increasingly, anywhere from several to thousands of networks can be created from brain imaging, gene co-expression data, or microbiome measurements. And these networks, in turn, are being looked to as potentially powerful features to be used in modeling. However, with networks being non-Euclidean in nature, how best to incorporate them into standard modeling tasks is not obvious. In this paper, we propose a Bayesian modeling framework that provides a unified approach to binary classification, anomaly detection, and survival analysis with network inputs. We encode the networks in the kernel of a Gaussian process prior via their pairwise differences and we discuss several choices of provably positive definite kernel that can be plugged into our models. Although our methods are widely applicable, we are motivated here in particular by microbiome research (where network analysis is emerging as the standard approach for capturing the interconnectedness of microbial taxa across both time and space) and its potential for reducing preterm delivery and improving personalization of prenatal care.

We produce a series of Central Limit Theorems (CLTs) associated to compact metric measure spaces $(K,d,\eta)$, with $\eta$ a reasonable probability measure. For the first CLT, we can ignore $\eta$ by isometrically embedding $K$ into ${\mathcal C}(K)$, the space of continuous functions on $K$ with the sup norm, and then applying known CLTs for sample means on Banach spaces (Theorem 3.1). However, the sample mean makes no sense back on $K$, so using $\eta$ we develop a CLT for the sample Fr\'echet mean (Corollary 4.1). This involves working on the closed convex hull of the embedded image of $K$. To work in the easier Hilbert space setting of $L^2(K,\eta)$, we have to modify the metric $d$ to a related metric $d_\eta$. We obtain an $L^2$-CLT for both the sample mean and the sample Fr\'echet mean (Theorem 5.1), and we relate the Fr\'echet sample and population means on the closed convex hull to the Fr\'echet means on the image of $K$. Since the $L^2$ and $L^\infty$ norms play important roles, in Section 6 we develop a metric-measure criterion relating $d$ and $\eta$ under which all $L^p$ norms are equivalent.

Gradient descent, or negative gradient flow, is a standard technique in optimization to find minima of functions. Many implementations of gradient descent rely on discretized versions, i.e., moving in the gradient direction for a set step size, recomputing the gradient, and continuing. In this paper, we present an approach to manifold learning where gradient descent takes place in the infinite dimensional space $\mathcal{E} = {\rm Emb}(M,\mathbb{R}^N)$ of smooth embeddings $\phi$ of a manifold $M$ into $\mathbb{R}^N$. Implementing a discretized version of gradient descent for $P:\mathcal{E}\to {\mathbb R}$, a penalty function that scores an embedding $\phi \in \mathcal{E}$, requires estimating how far we can move in a fixed direction -- the direction of one gradient step -- before leaving the space of smooth embeddings. Our main result is to give an explicit lower bound for this step length in terms of the Riemannian geometry of $\phi(M)$. In particular, we consider the case when the gradient of $P$ is pointwise normal to the embedded manifold $\phi(M)$. We prove this case arises when $P$ is invariant under diffeomorphisms of $M$, a natural condition in manifold learning.

It is becoming increasingly common to see large collections of network data objects -- that is, data sets in which a network is viewed as a fundamental unit of observation. As a result, there is a pressing need to develop network-based analogues of even many of the most basic tools already standard for scalar and vector data. In this paper, our focus is on averages of unlabeled, undirected networks with edge weights. Specifically, we (i) characterize a certain notion of the space of all such networks, (ii) describe key topological and geometric properties of this space relevant to doing probability and statistics thereupon, and (iii) use these properties to establish the asymptotic behavior of a generalized notion of an empirical mean under sampling from a distribution supported on this space. Our results rely on a combination of tools from geometry, probability theory, and statistical shape analysis. In particular, the lack of vertex labeling necessitates working with a quotient space modding out permutations of labels. This results in a nontrivial geometry for the space of unlabeled networks, which in turn is found to have important implications on the types of probabilistic and statistical results that may be obtained and the techniques needed to obtain them., Comment: This version contains two extensions of the main result on the uniqueness of the Fr\'echet mean, and has added references to related work

We propose a method to obtain optimal protocols for adiabatic ground-state preparation near the adiabatic limit, extending earlier ideas from [D. A. Sivak and G. E. Crooks, Phys. Rev. Lett. 108, 190602 (2012)] to quantum non-dissipative systems. The space of controllable parameters of isolated quantum many-body systems is endowed with a Riemannian quantum metric structure, which can be exploited when such systems are driven adiabatically. Here, we use this metric structure to construct optimal protocols in order to accomplish the task of adiabatic ground-state preparation in a fixed amount of time. Such optimal protocols are shown to be geodesics on the parameter manifold, maximizing the local fidelity. Physically, such protocols minimize the average energy fluctuations along the path. Our findings are illustrated on the Landau-Zener model and the anisotropic XY spin chain. In both cases we show that geodesic protocols drastically improve the final fidelity. Moreover, this happens even if one crosses a critical point, where the adiabatic perturbation theory fails., Comment: 5 pages, 2 figures + 4 pages supplement

We explore topological transitions in parameter space in order to enable adiabatic passages between regions adiabatically disconnected within a given parameter manifold. To this end, we study the Hamiltonian of two coupled qubits interacting with external magnetic fields, and make use of the analogy between the Berry curvature and magnetic fields in parameter space, with spectrum degeneracies associated to magnetic charges. Symmetry-breaking terms induce sharp topological transitions on these charge distributions, and we show how one can exploit this effect to bypass crossing degeneracies. We also investigate the curl of the Berry curvature, an interesting but as of yet not fully explored object, which together with its divergence uniquely defines this field. Finally, we suggest a simple method for measuring the Berry curvature, thereby showing how one can experimentally verify our results., Comment: 17 pages, 11 figures