A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Neural networks can represent complex piecewise functions efficiently.
We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the existence of a model for which the group acts preserving the geome…
BN refines local partition geometry in piecewise-affine networks during training.
In exchange for large quantities of data and processing power, deep neural networks have yielded models that provide state of the art predication capabilities in many fields. However, a lack of strong guarantees on their behaviour have raised concerns over their use in safety-critical applications. A first step to unde…
Paper presents ABGD for efficient piecewise linear regression in high dimensions.
Method identifies latent variables from high-dimensional data with piecewise affine mixing.
Paper develops algorithms for PWA systems with polynomial regret.
We study the coarse geometry of the moduli space of dilation tori with two singularities and the dynamical properties of the action of the Teichmuller flow on this moduli space. This leads to a proof that the vertical foliation of a dilation torus is almost always Morse-Smale. As a corollary, we get that the generic pi…
Max-affine regression method converges linearly using GD and SGD.
Study growth patterns in random networks using i.i.d. perturbations.
Paper proposes a new method for SP with covariates using PADR and ERM.
PAR provides a flexible framework for quantization in optimization problems.
This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give upper and lower bounds for the minimal number of triangles needed to forma triangulated embedded orientable surface in R^d having P as its geo…
Exact LAD line fitting via PALB with linear scaling and speed.
The paper designs neural networks with assurance for controlling nonlinear systems.
PARC uses piecewise linear predictors for regression and classification.
New method uses DC functions for piecewise linear regression.
Nonlinearity is crucial to the performance of a deep (neural) network (DN). To date there has been little progress understanding the menagerie of available nonlinearities, but recently progress has been made on understanding the rôle played by piecewise affine and convex nonlinearities like the ReLU and absolute value …
We consider the quasiconformal dilatation of projective transformations of the real projective plane. For non-affine transformations, the contour lines of dilatation form a hyperbolic pencil of circles, and these are the only circles that are mapped to circles. We apply this result to analyze the dilatation of the circ…
The paper provides results regarding the computational complexity of hybrid system identification. More precisely, we focus on the estimation of piecewise affine (PWA) maps from input-output data and analyze the complexity of computing a global minimizer of the error. Previous work showed that a global solution could b…
New EM algorithm improves deep generative network training.
Efficiently finds sparse solutions to max-plus equations for convex regression.
Normalizing flows attempt to model an arbitrary probability distribution through a set of invertible mappings. These transformations are required to achieve a tractable Jacobian determinant that can be used in high-dimensional scenarios. The first normalizing flow designs used coupling layer mappings built upon affine …
Discretizes Helfrich-type energies on surfaces using triangular complexes.
Quantum Monte Carlo speeds up option pricing for complex payoff functions.
BNN-DP improves robustness analysis of Bayesian Neural Networks.
This paper shows how to hedge financial risks with integer investments.
Batch normalization improves deep networks by aligning their decision boundaries with data.
We study the geometry of deep (neural) networks (DNs) with piecewise affine and convex nonlinearities. The layers of such DNs have been shown to be {\em max-affine spline operators} (MASOs) that partition their input space and apply a region-dependent affine mapping to their input to produce their output. We demonstrat…
This technical note extends recent results on the computational complexity of globally minimizing the error of piecewise-affine models to the related problem of minimizing the error of switching linear regression models. In particular, we show that, on the one hand the problem is NP-hard, but on the other hand, it admi…
New insights into Deep Autoencoders for better data approximation and generalization.
Subspace clustering refers to the problem of clustering high-dimensional data into a union of low-dimensional subspaces. Current subspace clustering approaches are usually based on a two-stage framework. In the first stage, an affinity matrix is generated from data. In the second one, spectral clustering is applied on …
Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and extend both of them using weighted lattices and a max- algebra with an ar…
A method for identifying NPWARX models with arbitrary domains using probabilistic mixture models.
We introduce a notion of measuring scales for quantum abelian gauge systems. At each measuring scale a finite dimensional affine space stores information about the evaluation of the curvature on a discrete family of surfaces. Affine maps from the spaces assigned to finer scales to those assigned to coarser scales play …
Two-dimensional affine A-nets in 3-space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The characterizing property of A-nets is planarity of vertex stars, so for generic A-nets the elementary quadrilaterals are skew. We classify the simply connected affine A-nets that can be ext…
I study Gromov-Hausdorff limits of complex curves endowed with singular flat metrics of constant diameter. I formulate a criterion that the limit is collapsed in terms of a certain piecewise affine weight function on the dual intersection complex of a semi-stable model of the degeneration introduced by Kontsevich and S…
A generalized semitoric system F:=(J,H): M --> R^2 on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S^1-action and is not necessarily proper. These systems can exhibit focus-focus singularities, which corresp…
A positive space is a space with a positive atlas, i.e. a collection of rational coordinate systems with subtraction free transition functions. The set of positive real points of a positive space is well defined. We define a tropical compactification of the latter. We show that it generalizes the Thurston compactificat…
New NN design for nonlinear systems control with guarantees.
This work generalizes bounds on the number of linear regions in CPWL NNs.
A celebrated theorem of Hadwiger states that the Euler-Poincaré characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely p…
We present new families of continuous piecewise linear (CPWL) functions in Rn having a number of affine pieces growing exponentially in . We show that these functions can be seen as the high-dimensional generalization of the triangle wave function used by Telgarsky in 2016. We prove that they can be computed by ReLU…
Duality principle for approximation of geometrical objects (also known as Eudoxus exhaustion method) was extended and perfected by Archimedes in his famous tractate "Measurement of circle". The main idea of the approximation method by Archimedes is to construct a sequence of pairs of inscribed and circumscribed polygon…
Kirigami-inspired math reveals shortest paths and ultimate shapes of cut paper.
The affine sphere construction gives, on any oriented surface, a one-to-one correspondence between convex -structures and holomorphic cubic differentials. Generalizing results of Benoist-Hulin, Loftin and Dumas-Wolf, we show that poles of order less than of cubic differentials correspond to finite vo…
Investigates stability of piecewise flat Ricci flow using analysis and simulations.
Theorem proves integrability for piecewise-smooth distributions.