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Steven Rosenberg

Publications and source records attributed to Steven Rosenberg.

At least 19 recordsLinked to original sources
arXiv2025

The Geometry of Loop Spaces V: Fundamental Groups of Geometric Transformation Groups

We use differential forms on loop spaces to prove that the fundamental group of certain geometric transformation groups is infinite. Examples include both finite and infinite dimensional Lie groups. The finite dimensional examples are the conformal group of $S^{4k+1}$ for a family of nonstandard metrics, and the group of pseudo-Hermitian transformations of a compact CR manifold. Infinite dimensional examples include the group of strict contact diffeomorphisms of a regular contact manifold, and other groups coming from symplectic and contact geometry.

math.DG↗
arXiv2025

Generalized Tangent Kernel: A Unified Geometric Foundation for Natural Gradient and Standard Gradient

Natural gradients have been widely studied from both theoretical and empirical perspectives, and it is commonly believed that natural gradients have advantages over standard (Euclidean) gradients in capturing the intrinsic geometric structure of the underlying function space and being invariant under reparameterization. However, for function optimization, a fundamental theoretical issue regarding the existence of natural gradients on the function space remains underexplored. We address this issue by providing a geometric perspective and mathematical framework for studying both natural gradient and standard gradient that is more complete than existing studies. The key tool that unifies natural gradient and standard gradient is a generalized form of the Neural Tangent Kernel (NTK), which we name the Generalized Tangent Kernel (GTK). Using a novel orthonormality property of GTK, we show that for a fixed parameterization, GTK determines a Riemannian metric on the entire function space which makes the standard gradient as "natural" as the natural gradient in capturing the intrinsic structure of the parameterized function space. Many aspects of this approach relate to RKHS theory. For the practical side of this theory paper, we showcase that our framework motivates new solutions to the non-immersion/degenerate case of natural gradient and leads to new families of natural/standard gradient descent methods.

cs.LG↗
arXiv2025

The Geometry of Loop Spaces IV: Closed Sasakian Manifolds

We prove that a closed regular $(4k+1)$-Sasakian manifold $(M,h_0)$ admits a family of non-isometric metrics $h_ρ, ρ\geq 0,$ such that $π_1({\rm Isom}(M, h_ρ))$, the fundamental group of the isometry group, is infinite for $ρ>0.$ For $M= S^{4k+1}$, this result holds for all $ρ>0$, but fails at $ρ=0.$

math.DG↗
arXiv2025

The Geometry of Loop Spaces II: Characteristic Classes

Using the Wodzicki residue, we build Wodzicki-Chern-Simons (WCS) classes in $H^{2k-1}(LM)$ associated to the residue Chern character on the loop space $LM$ of a Riemannian manifold $M^{2k-1}$. These WCS classes are associated to the $L^2$ connection and the Sobolev $s=1$ connections on $LM.$ The WCS classes detect several families of 5-manifolds whose isometry group has infinite fundamental group. These manifolds are the total spaces of the circle bundles associated to a multiple $pω, |p|\gg 0$, of the Kähler form $ω$ over an integral Kähler surface.

math.DG↗
arXiv2024

A Codimension Two Approach to the $\mathbb{S}^1$-Stability Conjecture

J. Rosenberg's $\mathbb{S}^1$-stability conjecture states that a closed oriented manifold $X$ admits a positive scalar curvature metric iff $X\times \mathbb{S}^1$ admits a positive scalar curvature metric $h$. As pointed out by J. Rosenberg and others, there are known counterexamples in dimension four. We prove this conjecture whenever $h$ satisfies a geometric bound which measures the discrepancy between $\partial_\theta\in T\mathbb{S}^1$ and the normal vector field to $X\times \{P\}$, for a fixed $P\in \mathbb{S}^1.$

math.DG↗
arXiv2024

The Geometry of Loop Spaces II: Corrections

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 $π_1({\rm Diff}(M))$ are replaced by results about $π_1({\rm Isom}(M))$, where Diff$(M)$, Isom$(M)$ refer to the diffeomorphism and isometry group of the manifold $M$.

math.DG↗
arXiv2024

Discretized Gradient Flow for Manifold Learning in the Space of Embeddings

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 $ϕ$ 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 $ϕ\in \mathcal{E}$, 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 $ϕ(M)$. In particular, we consider the case when the gradient of $P$ is pointwise normal to the embedded manifold $ϕ(M)$. We prove this case arises when $P$ is invariant under diffeomorphisms of $M$, a natural condition in manifold learning.

math.DG↗
arXiv2023

The Conformal Laplacian and Positive Scalar Curvature Metrics on Manifolds with Boundary

We give examples of spin $4$-manifolds with boundary $(M,\partial M)$ such that the boundary $\partial M$ has a positive scalar curvature metric which cannot be extended to a positive scalar curvature metric on $M$ with mean convex boundary. These manifolds have the equivalent analytic property that for any metric $g$ on $M$, the conformal Laplacian on $M$ with appropriate boundary conditions and the conformal Laplacian on $\partial M$ cannot both be positive. The obstruction to the positivity of the conformal Laplacians 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 directly treat metrics which are not a product near the boundary.

math.DG↗
arXiv2022

Occupant Plugload Management for Demand Response in Commercial Buildings: Field Experimentation and Statistical Characterization

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%

stat.AP↗
arXiv2021

Solving the Yamabe Problem by an Iterative Method on a Small Riemannian Domain

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.

math.DG↗
arXiv2021

Bayesian classification, anomaly detection, and survival analysis using network inputs with application to the microbiome

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.

stat.AP↗
arXiv2020

The Geometry of Loop Spaces III: Isometry Groups of Contact Manifolds

Let $M_p$ be a circle bundle with first Chern class $p[\omega]$ over a closed $4n$-dimensional integral symplectic manifold $\bigl(\overline{M},\omega\bigr)$. Equivalently, $M_p$ is a closed contact $(4n+1)$-manifold whose Reeb orbits are all closed and have the same period. For a metric $g$ on $M_p$ compatible with the symplectic structure and the geometry of the circle fiber, we use Wodzicki-Chern-Simons forms on the loop space $LM_p$ to prove that $\pi_1({\rm Isom}(M_p,g))$ is infinite for ${|p| \gg 0}$. We also give the first high-dimensional examples of nonvanishing Wodzicki-Pontryagin forms.

math.DG↗
arXiv2020

Central Limit Theorems on Compact Metric Spaces

We produce a series of Central Limit Theorems (CLTs) associated to compact metric measure spaces $(K,d,η)$, with $η$ a reasonable probability measure. For the first CLT, we can ignore $η$ 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 $η$ we develop a CLT for the sample Fréchet 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 $L^2(K,η)$, we have to modify the metric $d$ to a related metric $d_η$. We obtain an $L^2$-CLT for both the sample mean and the sample Fréchet mean (Theorem 5.1), and we relate the Fréchet sample and population means on the closed convex hull to the Fréchet 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 $η$ under which all $L^p$ norms are equivalent.

math.PR↗
arXiv2019

Averages of Unlabeled Networks: Geometric Characterization and Asymptotic Behavior

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.

math.ST↗
arXiv2017

Hypothesis Testing For Network Data in Functional Neuroimaging

In recent years, it has become common practice in neuroscience to use networks to summarize relational information in a set of measurements, typically assumed to be reflective of either functional or structural relationships between regions of interest in the brain. One of the most basic tasks of interest in the analysis of such data is the testing of hypotheses, in answer to questions such as "Is there a difference between the networks of these two groups of subjects?" In the classical setting, where the unit of interest is a scalar or a vector, such questions are answered through the use of familiar two-sample testing strategies. Networks, however, are not Euclidean objects, and hence classical methods do not directly apply. We address this challenge by drawing on concepts and techniques from geometry, and high-dimensional statistical inference. Our work is based on a precise geometric characterization of the space of graph Laplacian matrices and a nonparametric notion of averaging due to Fréchet. We motivate and illustrate our resulting methodologies for testing in the context of networks derived from functional neuroimaging data on human subjects from the 1000 Functional Connectomes Project. In particular, we show that this global test is more statistical powerful, than a mass-univariate approach. In addition, we have also provided a method for visualizing the individual contribution of each edge to the overall test statistic.

stat.AP↗
arXiv2016

Geodesic Paths for Quantum Many-Body Systems

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.

cond-mat.quant-gas↗
arXiv2016

Enabling Adiabatic Passages Between Disjoint Regions in Parameter Space through Topological Transitions

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.

cond-mat.mes-hall↗