Related papers: Disentangling a Deep Learned Volume Formula
Freedman, Kitaev, and Wang [arXiv:quant-ph/0001071], and later Aharonov, Jones, and Landau [arXiv:quant-ph/0511096], established a quantum algorithm to "additively" approximate the Jones polynomial V(L,t) at any principal root of unity t.…
In this article, we give a rough, and so not complete yet, proof of Kashaev's conjecture, that is, the volume conjecture for hyperbolic knots, where the hyperbolicity equations associated to knot diagrams appear as the stationary phase…
We consider one loop quantum corrections to soliton mass for the ${\cal N}=1$ supersymmetric extension of the (1+1)-dimensional scalar field theory with the potential $U(\phi) = \phi^2 \cos^2\left(\ln \phi^2\right)$. First, we compute the…
We renormalize the Chern-Simons invariant for convex-cocompact hyperbolic 3-manifolds by finding the asymptotics along an equidistance foliation. We prove that the metric Chern-Simons invariant has an exponentially divergent term given by…
An algorithm of the tensor renormalization group is proposed based on a randomized algorithm for singular value decomposition. Our algorithm is applicable to a broad range of two-dimensional classical models. In the case of a square…
Using gravitational waves to probe the geometry of the ringing remnant black hole formed in a binary black hole coalescence is a well-established way to test Einstein's theory of general relativity. However, doing so requires knowledge of…
Universal approximation theory offers a foundational framework to verify neural network expressiveness, enabling principled utilization in real-world applications. However, most existing theoretical constructions are established by…
This paper presents a saddlepoint approximation of the random-coding union bound of Polyanskiy et al. for i.i.d. random coding over discrete memoryless channels. The approximation is single-letter, and can thus be computed efficiently.…
We show that for a large class of hyperbolic knots and links, we can determine bounds on the volume of the link complement from combinatorial information given by a link diagram. Specifically, there is a universal constant C such that if a…
Knot and link polynomials are topological invariants calculated from the expectation value of loop operators in topological field theories. In 3D Chern-Simons theory, these invariants can be found from crossing and braiding matrices of…
We examine the necessary and sufficient complexity of neural networks to approximate functions from different smoothness spaces under the restriction of encodable network weights. Based on an entropy argument, we start by proving lower…
We derive rigorous bounds on the error resulting from the approximation of the solution of parametric hyperbolic scalar conservation laws with ReLU neural networks. We show that the approximation error can be made as small as desired with…
We develop a constructive piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The main ingredients to derive optimal error estimates for an averaged Taylor polynomial are…
We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced…
We study approximations of polytopes in the standard model for computing polytopes using Minkowski sums and (convex hulls of) unions. Specifically, we study the ability to approximate a target polytope by polytopes of a given depth. Our…
Based on the data of 12-17-crossing knots, we establish three new conjectures about the hyperbolic volume and knot cohomology: (1) There exists a constant $a \in R_{>0}$ such that the percentage of knots for which the following inequality…
We study three-dimensional Chern-Simons theory with complex gauge group SL(2,C), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single…
We provide exact integral formulas for hyperbolic and spherical volumes of cone-manifolds whose underlying space is the $3$-sphere and whose singular set belongs to three infinite families of two-bridge knots: $C(2n,2)$ (twist knots),…
We study the improper learning of multi-layer neural networks. Suppose that the neural network to be learned has $k$ hidden layers and that the $\ell_1$-norm of the incoming weights of any neuron is bounded by $L$. We present a kernel-based…
We give a volume formula of hyperbolic knot complements using twisted Alexander invariants.