arXiv subjects

Jack Kendrick

Publications and source records attributed to Jack Kendrick.

7 recordsLinked to original sources

arXiv2025

The Tree-SNE Tree Exists

The clustering and visualisation of high-dimensional data is a ubiquitous task in modern data science. Popular techniques include nonlinear dimensionality reduction methods like t-SNE or UMAP. These methods face the `scale-problem' of clustering: when dealing with the MNIST dataset, do we want to distinguish different digits or do we want to distinguish different ways of writing the digits? The answer is task dependent and depends on scale. We revisit an idea of Robinson & Pierce-Hoffman that exploits an underlying scaling symmetry in t-SNE to replace 2-dimensional with (2+1)-dimensional embeddings where the additional parameter accounts for scale. This gives rise to the t-SNE tree (short: tree-SNE). We prove that the optimal embedding depends continuously on the scaling parameter for all initial conditions outside a set of measure 0: the tree-SNE tree exists. This idea conceivably extends to other attraction-repulsion methods and is illustrated on several examples.

stat.ML↗

arXiv2025

Universal Gr\"obner Bases of (Universal) Multiview Ideals

Multiview ideals arise from the geometry of image formation in pinhole cameras, and universal multiview ideals are their analogs for unknown cameras. We prove that a natural collection of polynomials form a universal Gr\"obner basis for both types of ideals using a criterion introduced by Huang and Larson, and include a proof of their criterion in our setting. Symmetry reduction and induction enable the method to be deployed on an infinite family of ideals. We also give an explicit description of the matroids on which the methodology depends, in the context of multiview ideals.

math.AC↗

arXiv2025

Identifying Orbit Lengths for Promotion

In this work we study Sch\"utzenberger's promotion operator on standard Young tableaux via a corresponding graphical construction known as $m-$diagrams. In particular, we prove that certain internal structures of SYT are preserved under promotion and correspond to distinct components of $m-$diagrams. By treating these structures as atomic parts of the $m-$diagram, we provide a simple algorithm for computing the promotion orbit length of rectangular SYT. We conclude the paper by applying our results to (column) semi-standard Young tableaux and prove a formula for the promotion orbit lengths of rectangular (column) SSYT.

math.CO↗

arXiv2025

Density Estimation on Rectifiable Sets

Kernel density estimation is a popular method for estimating unseen probability distributions. However, the convergence of these classical estimators to the true density slows down in high dimensions. Moreover, they do not define meaningful probability distributions when the intrinsic dimension of data is much smaller than its ambient dimension. We build on previous work on density estimation on manifolds to show that a modified kernel density estimator converges to the true density on $d-$rectifiable sets. As a special case, we consider algebraic varieties and semi-algebraic sets and prove a convergence rate in this setting. We conclude the paper with a numerical experiment illustrating the convergence of this estimator on sparse data.

math.ST↗

arXiv2025

Brackets and Projective Geometry in Macaulay2

We introduce the Brackets package for the computer algebra system Macaulay2, which provides convenient syntax for computations involving the classical invariants of the special linear group. We describe our implementation of bracket rings and Grassmann-Cayley algebras, and illustrate basic functionality such as the straightening algorithm on examples from projective and enumerative geometry.

math.AG↗

arXiv2025

Invariant Kernels: Rank Stabilization and Generalization Across Dimensions

Symmetry arises often when learning from high dimensional data. For example, data sets consisting of point clouds, graphs, and unordered sets appear routinely in contemporary applications, and exhibit rich underlying symmetries. Understanding the benefits of symmetry on the statistical and numerical efficiency of learning algorithms is an active area of research. In this work, we show that symmetry has a pronounced impact on the rank of kernel matrices. Specifically, we compute the rank of a polynomial kernel of fixed degree that is invariant under various groups acting independently on its two arguments. In concrete circumstances, including the three aforementioned examples, symmetry dramatically decreases the rank making it independent of the data dimension. In such settings, we show that a simple regression procedure is minimax optimal for estimating an invariant polynomial from finitely many samples drawn across different dimensions. We complete the paper with numerical experiments that illustrate our findings.

math.OC↗

arXiv2021

Dihedral Linking Invariants

A Fox p-colored knot $K$ in $S^3$ gives rise to a corresponding $p$-fold dihedral branched cover $M$ of $S^3$ along $K$. The pre-image of the knot $K$ under the covering map is a $\dfrac{p+1}{2}$-component link $L$ in $M$, and the set of pairwise linking numbers of the components of $L$ is an invariant of $K$. This powerful invariant played a key role in the development of early knot tables, and appears in formulas for many other important knot and manifold invariants. We give an algorithm for computing this invariant for all odd $p$, generalizing an algorithm of Perko. We then extend this algorithm to compute linking numbers of arbitrary curves in a $p$-fold dihedral branched cover of $S^3$ along $K$. As an application, we compute Kjuchukova's ribbon obstruction $\Xi_p$ using a method of the first author and Kjuchukova. We also tabulate the dihedral linking invariant for all $p$-colorings of prime knots of crossing number less than or equal to 13, with $p\geq 3$ prime. Finally, we demonstrate the strength of the dihedral linking invariant by comparing it to several polynomial invariants. For example, the dihedral linking invariant distinguishes more than 98% of the 1183 prime non-mutant knot pairs with the same Fox coloring invariant and the same HOMFLY-PT polynomial through 13 crossings.

math.GT↗