arXiv subjects
Publications and source records attributed to Leonard Schmitz.
We introduce SignatureTensors.jl, a new package for computing signature tensors of paths in julia. We present its core functionality and demonstrate its use through illustrative examples. The package is compatible with the computer algebra system OSCAR, enabling both exact and numerical computations with signatures.
We present a new algorithm for recovering paths from their third-order signature tensors, an inverse problem in rough analysis. Our algorithm provides the exact solution to this recovery problem and improves upon current approaches by an order of magnitude. It relies on generalized normal forms and stabilizers of group actions via matrix-tensor congruence. We apply randomized transformation techniques that avoid the task of solving nonlinear polynomial systems associated to degenerate paths, and accompany our methods with an efficient implementation in the computer algebra system OSCAR.
Splines are central objects for the interpolation of discrete data via piecewise smooth paths. Their iterated-integral signature is an infinite collection of tensors which characterizes paths almost uniquely. We study truncations of this collection, which define algebraic maps from parameter space to tensor space. We prove that the images of these maps are given by orbits of a matrix-tensor action. Furthermore, taking the Zariski closure, we define and study varieties of spline signature tensors. We determine dimension and degree of these tensor varieties in a number of examples, relying on symbolic computations. With a view towards learning, constructing paths with a given signature tensor translates to studying the fibers of the signature map. We use computational methods to determine their cardinality, with a focus on its dependence on different classes of splines. We observe in explicit examples that reconstructing splines from a given signature tensor of a path yields close approximations of the original path.
The expected signature of a family of paths need not be a signature of a path itself. Motivated by this, we consider the notion of a Lie group barycenter introduced by Buser and Karcher to propose a barycenter on path signatures. We show that every element of the free nilpotent Lie group is a barycenter of a group sample, where all but one sample element can be fixed arbitrarily. In the case of piecewise linear paths, we study the problem of recovering an underlying path corresponding to the barycenter of signatures. We determine the minimal number of segments required to learn from signature matrices, providing explicit transformations to the associated congruence normal forms.
The signature of a membrane is a sequence of tensors whose entries are iterated integrals. We study algebraic properties of membrane signatures, with a focus on signature matrices of polynomial and piecewise bilinear membranes. Generalizing known results for path signatures, we show that the two families of membranes admit the same set of signature matrices and we examine the corresponding affine varieties. In particular, we prove that there are no algebraic relations on signature matrices of membranes, in contrast to the path case. We complement our results by a linear time algorithm for the computation of signature tensors for piecewise bilinear membranes.
Data in the form of images or higher-order tensors is ubiquitous in modern deep learning applications. Owing to their inherent high dimensionality, the need for subquadratic layers processing such data is even more pressing than for sequence data. We propose a novel tensor-to-tensor layer with linear cost in the input size, utilizing the mathematical gadget of ``corner trees'' from the field of permutation counting. In particular, for order-two tensors, we provide an image-to-image layer that can be plugged into image processing pipelines. On the one hand, our method can be seen as a higher-order generalization of state-space models. On the other hand, it is based on a multiparameter generalization of the signature of iterated integrals (or sums). The proposed tensor-to-tensor concept is used to build a neural network layer called the Fast Iterated Sums (FIS) layer which integrates seamlessly with other layer types. We demonstrate the usability of the FIS layer with both classification and anomaly detection tasks. By replacing some layers of a smaller ResNet architecture with FIS, a similar accuracy (with a difference of only 0.1\%) was achieved in comparison to a larger ResNet while reducing the number of trainable parameters and multi-add operations. The FIS layer was also used to build an anomaly detection model that achieved an average AUROC of 97.3\% on the texture images of the popular MVTec AD dataset. The processing and modelling codes are publicly available at https://github.com/diehlj/fast-iterated-sums.
Non-commutative Gröbner bases of two-sided ideals are not necessarily finite. Motivated by this, we provide a closed-form description of a finite and reduced Gröbner bases for the two-sided ideal used in the construction of Wangs quantum symmetric group. In particular, this proves that the word problem for quantum symmetric groups is decidable.
Quasisymmetric functions have recently been used in time series analysis as polynomial features that are invariant under, so-called, dynamic time warping. We extend this notion to data indexed by two parameters and thus provide warping invariants for images. We show that two-parameter quasisymmetric functions are complete in a certain sense, and provide a two-parameter quasi-shuffle identity. A compatible coproduct is based on diagonal concatenation of the input data, leading to a (weak) form of Chen's identity.
Signature transforms have recently been extended to data indexed by two and more parameters. With free Lyndon generators, ideas from $\mathbf{B}_\infty$-algebras and a novel two-parameter Hoffman exponential, we provide three classes of isomorphisms between the underlying two-parameter shuffle and quasi-shuffle algebras. In particular, we provide a Hopf algebraic connection to the (classical, one-parameter) shuffle algebra over the extended alphabet of connected matrix compositions.
We establish the well-definedness of the barycenter (in the sense of Buser and Karcher) for every integrable measure on the free nilpotent Lie group of step $L$ (over $\mathbb{R}^d$). We provide two algorithms for computing it, using methods from Lie theory (namely, the Baker-Campbell-Hausdorff formula) and from the theory of Gröbner bases of modules. Our main motivation stems from measures induced by iterated-integrals signatures, and we calculate the barycenter for the signature of the Brownian motion.