arXiv research

This dissertation explores the integration of learning and analogy-making through the development of a computer program, called Analogator, that learns to make analogies by example. By "seeing" many different analogy problems, along with possible solutions, Analogator gradually develops an ability to make new analogies…

Analog method solves portfolio optimization problems faster and more efficiently.

Fault-tolerant neural networks inspired by biological error correction codes.

Defines an odd analog of Plamenevskaya's invariant for transverse links.

Analog arrays are a promising upcoming hardware technology with the potential to drastically speed up deep learning. Their main advantage is that they compute matrix-vector products in constant time, irrespective of the size of the matrix. However, early convolution layers in ConvNets map very unfavorably onto analog a…

Analog deep learning shows promise but faces scalability challenges.

The Spencer cohomology of certain graded Lie superalgebras are completely computed. This cohomology is interpreted as analogs of Riemann and Penrose tensors on supermanifolds. The results make it manifest that there is no simple generalization of Borel-Weil-Bott's theorem for Lie superalgebras.

Analogy-making is a key method for computer algorithms to generate both natural and creative music pieces. In general, an analogy is made by partially transferring the music abstractions, i.e., high-level representations and their relationships, from one piece to another; however, this procedure requires disentangling …

Neuromorphic Computing is a nascent research field in which models and devices are designed to process information by emulating biological neural systems. Thanks to their superior energy efficiency, analog neuromorphic systems are highly promising for embedded, wearable, and implantable systems. However, optimizing neu…

New method uses surrogate gradients to train efficient spiking networks on neuromorphic hardware.

The availability of large idea repositories (e.g., the U.S. patent database) could significantly accelerate innovation and discovery by providing people with inspiration from solutions to analogous problems. However, finding useful analogies in these large, messy, real-world repositories remains a persistent challenge …

We explore the robustness of recurrent neural networks when the computations within the network are noisy. One of the motivations for looking into this problem is to reduce the high power cost of conventional computing of neural network operations through the use of analog neuromorphic circuits. Traditional GPU/CPU-cen…

We constructed an analog electrical circuit which generates fluctuations in which probability density function has power law tails. In the circuit fluctuations with an arbitrary exponent of the power law can be obtained by adjusting the resistance. With this low cost circuit the random fluctuations which have the simil…

The success of deep learning has brought forth a wave of interest in computer hardware design to better meet the high demands of neural network inference. In particular, analog computing hardware has been heavily motivated specifically for accelerating neural networks, based on either electronic, optical or photonic de…

Novel algorithm for privacy-preserving distributed learning in analog domain.

Bayesian optimization with Gaussian process as surrogate model has been successfully applied to analog circuit synthesis. In the traditional Gaussian process regression model, the kernel functions are defined explicitly. The computational complexity of training is O(N 3 ), and the computation complexity of prediction i…

I survey methods from differential geometry, algebraic geometry and representation theory relevant for the permanent v. determinant problem from computer science, an algebraic analog of the P v. NP problem.

A trend towards energy-efficiency, security and privacy has led to a recent focus on deploying DNNs on microcontrollers. However, limits on compute and memory resources restrict the size and the complexity of the ML models deployable in these systems. Computation-In-Memory architectures based on resistive nonvolatile m…

We study two aspects of noisy computations during inference. The first aspect is how to mitigate their side effects for naturally trained deep learning systems. One of the motivations for looking into this problem is to reduce the high power cost of conventional computing of neural networks through the use of analog ne…

How spiking networks are able to perform probabilistic inference is an intriguing question, not only for understanding information processing in the brain, but also for transferring these computational principles to neuromorphic silicon circuits. A number of computationally powerful spiking network models have been pro…

Analog forecasting uses local dynamics to predict chaotic systems.

Computes mapping class groups of 4-manifolds with boundary.

New formula for dual knots using involutions.

ADR helps LLMs find and use historical analogies for foresight analysis.

Analog methods improve forecast accuracy in complex models.

We obtain formulas for the first and second cohomology groups of a general current Lie algebra with coefficients in the "current" module, and apply them to compute structure functions for manifolds of loops with values in compact Hermitian symmetric spaces.

The paper evaluates the probability distributions of analog-to-target distances for multiple analogs.

We investigate the analogy between the large N expansion in normal matrix models and the asymptotic expansion of the determinant of the Hilb map, appearing in the study of critical metrics on complex manifolds via projective embeddings. This analogy helps to understand the geometric meaning of the expansion of matrix m…

Lattice cohomology, defined by Némethi in (arXiv:0709.0841), is an invariant of negative definite plumbed 3-manifolds which conjecturally computes the Heegaard Floer homology HF^+. We prove a surgery exact triangle for the lattice cohomology analogous to the one for HF^+. This is a step towards comparing these two inva…

Approximates smooth surfaces using Laguerre geometry meshes.

Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth $C^1$-surface built from surface patches of Dupin cyclides, each patch being bounded by curvature lines of the supporting cyclide. An explicit de…

Approximates surfaces using Laguerre geometry with spherical faces.

Spiking neural networks (SNNs) are brain-inspired mathematical models with the ability to process information in the form of spikes. SNNs are expected to provide not only new machine-learning algorithms, but also energy-efficient computational models when implemented in VLSI circuits. In this paper, we propose a novel …

We formulate and prove an analog of the Hopf Index Theorem for Riemannian foliations. We compute the basic Euler characteristic of a closed Riemannian manifold as a sum of indices of a non-degenerate basic vector field at critical leaf closures. The primary tool used to establish this result is an adaptation to foliati…

We prove in a simple and coordinate-free way the equivalence bteween the classical definitions of the mass or the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asympto…

In this paper, we review recent work published over the last 3 years under the umbrella of Neuromorphic engineering to analyze what are the common features among such systems. We see that there is no clear consensus but each system has one or more of the following features:(1) Analog computing (2) Non vonNeumann Archit…

Building on a specific formalization of analogical relationships of the form "A relates to B as C relates to D", we establish a connection between two important subfields of artificial intelligence, namely analogical reasoning and kernel-based machine learning. More specifically, we show that so-called analogical propo…

Computes Vafa-Witten invariants of 3-manifolds.

Computes derivatives of sections in vector bundles using Lie derivatives.

Researchers compute c-projective symmetry algebras for Kähler surfaces.

We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkahler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to q…

Fix a finite group $G$. We analyze the computational complexity of the problem of counting homomorphisms $π_1(X) \to G$, where $X$ is a topological space treated as computational input. We are especially interested in requiring $G$ to be a fixed, finite, nonabelian, simple group. We then consider two cases: when the in…

We present a graph manifold analog of the Jankins-Neumann classification of Seifert fibered spaces over $S^2$ admitting taut foliations, providing a finite recursive formula to compute the L-space Dehn-filling interval for any graph manifold with torus boundary. As an application of a generalization of this result to F…

We show that the recently introduced L1TV functional can be used to explicitly compute the flat norm for co-dimension one boundaries. While this observation alone is very useful, other important implications for image analysis and shape statistics include a method for denoising sets which are not boundaries or which ha…

This is an expository article of our work on analogies between knot theory and algebraic number theory. We shall discuss foundational analogies between knots and primes, 3-manifolds and number rings mainly from the group-theoretic point of view.

We describe a hyperbolic version of the Ambartzumian-Pleijel identity. We use this identity to prove the hyperbolic Crofton formula and the hyperbolic isoperimetric inequality. This identity also provides a way to compute the chord length distribution for an ideal polygon in the hyperbolic plane. The analogous results …

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using parti…

Niebrzydowski tribrackets are ternary operations on sets satisfying conditions obtained from the oriented Reidemeister moves such that the set of tribracket colorings of an oriented knot or link diagram is an invariant of oriented knots and links. We introduce tribracket modules analogous to quandle/biquandle/rack modu…