arXiv subjects

Joe Kileel

Publications and source records attributed to Joe Kileel.

At least 19 recordsLinked to original sources

arXiv2026

Group Invariant Spectral Embedding

Spectral embedding methods are widely used for dimensionality reduction and clustering of high-dimensional datasets with intrinsic low-dimensional structures. Although many datasets of practical interest exhibit invariance under symmetries such as rotations, standard spectral embedding methods do not account for this, treating symmetry-related data points as unrelated. Our approach to this problem is to incorporate the symmetries directly into the affinity kernels used for spectral embedding. We analyze the case of a Riemannian data manifold $M$ with symmetries given by a compact Lie group~$G$ and prove that, under suitable conditions, graph Laplacians constructed from three types of invariant kernels converge pointwise to explicit second-order differential operators on the quotient space $M/G$. Our analysis implies improved convergence rates, as the effective dimension drops according to the dimension of the group. We validate our approach on datasets with $\mathrm{SO}(2)$ or $\mathrm{SO}(3)$ symmetry, and show that $G$-invariant spectral embedding recovers the intrinsic geometry of the data, in contrast to standard spectral embedding, which fails to do so even in the limit of infinite data.

cs.LG↗

arXiv2026

Testing the mixture model hypothesis via spectral gap

In this paper, we study the problem of testing whether or not a given probability measure $\mu$ on $\mathbb{R}^{d}$ can be decomposed as a mixture of two probability measures whose second order statistics are significantly different. We call this the problem of testing the mixture model hypothesis. To tackle it, we introduce a new set of computable orthogonal invariants of $\mu$, namely, the eigenvalues of the 4th moment operator $T_{\mu}$ associated with the measure. We prove that the largest eigenvalue is always an outlier eigenvalue. Further, we show how the first and second largest eigenvalues of $T_{\mu}$ give nonasymptotic bounds for this problem and give a complete resolution of the asymptotic version of the problem under the $L^{8}$-$L^{2}$ equivalence assumption.

math.PR↗

arXiv2026

QuadSync: Quadrifocal Tensor Synchronization via Tucker Decomposition

In structure from motion, quadrifocal tensors capture more information than their pairwise counterparts (essential matrices), yet they have often been thought of as impractical and only of theoretical interest. In this work, we challenge such beliefs by providing a new framework to recover $n$ cameras from the corresponding collection of quadrifocal tensors. We form the block quadrifocal tensor and show that it admits a Tucker decomposition whose factor matrices are the stacked camera matrices, and which thus has a multilinear rank of (4,~4,~4,~4) independent of $n$. We develop the first synchronization algorithm for quadrifocal tensors, using Tucker decomposition, alternating direction method of multipliers, and iteratively reweighted least squares. We further establish relationships between the block quadrifocal, trifocal, and bifocal tensors, and introduce an algorithm that jointly synchronizes these three entities. Numerical experiments demonstrate the effectiveness of our methods on modern datasets, indicating the potential and importance of using higher-order information in synchronization.

cs.CV↗

arXiv2026

First Proof

To assess the ability of current AI systems to correctly answer research-level mathematics questions, we share a set of ten math questions which have arisen naturally in the research process of the authors. The questions had not been shared publicly until now; the answers are known to the authors of the questions but will remain encrypted for a short time.

cs.AI↗

arXiv2026

Multi-context principal component analysis

Principal component analysis (PCA) is a tool to capture factors that explain variation in data. Across domains, data are now collected across multiple contexts (for example, individuals with different diseases, cells of different types, or words across texts). While the factors explaining variation in data are undoubtedly shared across subsets of contexts, no tools currently exist to systematically recover such factors. We develop multi-context principal component analysis (MCPCA), a theoretical and algorithmic framework that decomposes data into factors shared across subsets of contexts. Applied to gene expression, MCPCA reveals axes of variation shared across subsets of cancer types and an axis whose variability in tumor cells, but not mean, is associated with lung cancer progression. Applied to contextualized word embeddings from language models, MCPCA maps stages of a debate on human nature, revealing a discussion between science and fiction over decades. These axes are not found by combining data across contexts or by restricting to individual contexts. MCPCA is a principled generalization of PCA to address the challenge of understanding factors underlying data across contexts.

stat.ML↗

arXiv2025

Two Datasets Are Better Than One: Method of Double Moments for 3-D Reconstruction in Cryo-EM

Cryo-electron microscopy (cryo-EM) is a powerful imaging technique for reconstructing three-dimensional molecular structures from noisy tomographic projection images of randomly oriented particles. We introduce a new data fusion framework, termed the method of double moments (MoDM), which reconstructs molecular structures from two instances of the second-order moment of projection images obtained under distinct orientation distributions: one uniform, the other non-uniform and unknown. We prove that these moments generically uniquely determine the underlying structure, up to a global rotation and reflection, and we develop a convex-relaxation-based algorithm that achieves accurate recovery using only second-order statistics. Our results demonstrate the advantage of collecting and modeling multiple datasets under different experimental conditions, illustrating that leveraging dataset diversity can substantially enhance reconstruction quality in computational imaging tasks.

cs.CV↗

arXiv2025

Multi-subspace power method for decomposing partially symmetric tensors

We present an algorithm for low rank decomposition of tensors of any symmetry type, from fully asymmetric to fully symmetric. It recovers the decomposition one summand at a time via the higher-order power method. This approach is known to fail in general: there need not be a relationship between the summands of a decomposition and the (partially symmetric) singular vector tuples (pSVTs) of the tensor. Our approach overcomes this problem by transforming the input to a tensor with orthonormal slices, via orthogonalization of a flattening. The summands of the decomposition of the original tensor can be recovered from the pSVTs of this new transformed tensor. We introduce a shifted power method for computing pSVTs and prove its global convergence. Numerical experiments demonstrate that our algorithm achieves higher accuracy and faster runtime than existing methods.

math.NA↗

arXiv2025

SO(3)-invariant PCA with application to molecular data

Principal component analysis (PCA) is a fundamental technique for dimensionality reduction and denoising; however, its application to three-dimensional data with arbitrary orientations -- common in structural biology -- presents significant challenges. A naive approach requires augmenting the dataset with many rotated copies of each sample, incurring prohibitive computational costs. In this paper, we extend PCA to 3D volumetric datasets with unknown orientations by developing an efficient and principled framework for SO(3)-invariant PCA that implicitly accounts for all rotations without explicit data augmentation. By exploiting underlying algebraic structure, we demonstrate that the computation involves only the square root of the total number of covariance entries, resulting in a substantial reduction in complexity. We validate the method on real-world molecular datasets, demonstrating its effectiveness and opening up new possibilities for large-scale, high-dimensional reconstruction problems.

eess.SP↗

arXiv2025

Efficient Tensor Decomposition via Moment Matrix Extension

Motivated by a flurry of recent work on efficient tensor decomposition algorithms, we show that the celebrated moment matrix extension algorithm of Brachat, Comon, Mourrain, and Tsigaridas for symmetric tensor canonical polyadic (CP) decomposition can be made efficient under the right conditions. We first show that the crucial property determining the complexity of the algorithm is the regularity of a target decomposition. This allows us to reduce the complexity of the vanilla algorithm, while also unifying results from previous works. We then show that for tensors in $S^d\mathbb{C}^{n+1}$ with $d$ even, low enough regularity can reduce finding a symmetric tensor decomposition to solving a system of linear equations. For order-$4$ tensors we prove that generic tensors of rank up to $r=2n+1$ can be decomposed efficiently via moment matrix extension, exceeding the rank threshold allowed by simultaneous diagonalization. We then formulate a conjecture that states for generic order-$4$ tensors of rank $r=O(n^2)$ the induced linear system is sufficient for efficient tensor decomposition, matching the asymptotics of existing algorithms and in fact improving the leading coefficient. Towards this conjecture we give computer assisted proofs that the statement holds for $n=2, \dots, 17$. Next we demonstrate that classes of nonidentifiable tensors can be decomposed efficiently via the moment matrix extension algorithm, bypassing the usual need for uniqueness of decomposition. Of particular interest is the class of monomials, for which the extension algorithm is not only efficient but also improves on existing theory by explicitly parameterizing the space of decompositions. Code for implementations of the efficient algorithm for generic tensors and monomials are provided, along with several numerical examples.

math.AG↗

arXiv2025

Norming Sets for Tensor and Polynomial Sketching

This paper develops the sketching (i.e., randomized dimension reduction) theory for real algebraic varieties and images of polynomial maps, including, e.g., the set of low rank tensors and tensor networks. Through the lens of norming sets, we provide a framework for controlling the sketching dimension for \textit{any} sketch operator used to embed said sets, including sub-Gaussian, fast Johnson-Lindenstrauss, and tensor structured sketch operators. Leveraging norming set theory, we propose a new sketching method called the median sketch. It embeds such a set $V$ using only $\widetilde{\mathcal{O}}(\dim V)$ tensor structured or sparse linear measurements.

math.NA↗

arXiv2025

Higher-Order Group Synchronization

Group synchronization is the problem of determining reliable global estimates from noisy local measurements on networks. The typical task for group synchronization is to assign elements of a group to the nodes of a graph in a way that respects group elements given on the edges which encode information about local pairwise relationships between the nodes. In this paper, we introduce a novel higher-order group synchronization problem which operates on a hypergraph and seeks to synchronize higher-order local measurements on the hyperedges to obtain global estimates on the nodes. Higher-order group synchronization is motivated by applications to computer vision and image processing, among other computational problems. First, we define the problem of higher-order group synchronization and discuss its mathematical foundations. Specifically, we give necessary and sufficient synchronizability conditions which establish the importance of cycle consistency in higher-order group synchronization. Then, we propose the first computational framework for general higher-order group synchronization; it acts globally and directly on higher-order measurements using a message passing algorithm. We discuss theoretical guarantees for our framework, including convergence analyses under outliers and noise. Finally, we show potential advantages of our method through numerical experiments. In particular, we show that in certain cases our higher-order method applied to rotational and angular synchronization outperforms standard pairwise synchronization methods and is more robust to outliers. We also show that our method has comparable performance on simulated cryo-electron microscopy (cryo-EM) data compared to a standard cryo-EM reconstruction package.

stat.ML↗

arXiv2025

The Fascinating World of 2 $\times$ 2 $\times$ 2 Tensors: Its Geometry and Optimization Challenges

This educational article highlights the geometric and algebraic complexities that distinguish tensors from matrices, to supplement coverage in advanced courses on linear algebra, matrix analysis, and tensor decompositions. Using the case of real-valued 2 $\times$ 2 $\times$ 2 tensors, we show how tensors violate many well-known properties of matrices: (1) The rank of a matrix is bounded by its smallest dimension, but a 2 $\times$ 2 $\times$ 2 tensor can be rank 3. (2) Matrices have a single typical rank, but the rank of a generic 2 $\times$ 2 $\times$ 2 tensor can be 2 or 3 - it has two typical ranks. (3) Any limit point of a sequence of matrices of rank $r$ is at most rank $r$, but a limit point of a sequence of 2 $\times$ 2 $\times$ 2 tensors of rank 2 can be rank 3 (a higher rank). (4) Matrices always have a best rank-$r$ approximation, but no rank-3 tensor of size 2 $\times$ 2 $\times$ 2 has a best rank-2 approximation. We unify the analysis of the matrix and tensor cases using tools from algebraic geometry and optimization, providing derivations of these surprising facts. To build intuition for the geometry of rank-constrained sets, students and educators can explore the geometry of matrix and tensor ranks via our interactive visualization tool.

math.NA↗

arXiv2025

Geometry and Optimization of Shallow Polynomial Networks

We study shallow neural networks with monomial activations and output dimension one. The function space for these models can be identified with a set of symmetric tensors with bounded rank. We describe general features of these networks, focusing on the relationship between width and optimization. We then consider teacher-student problems, which can be viewed as problems of low-rank tensor approximation with respect to non-standard inner products that are induced by the data distribution. In this setting, we introduce a teacher-metric data discriminant which encodes the qualitative behavior of the optimization as a function of the training data distribution. Finally, we focus on networks with quadratic activations, presenting an in-depth analysis of the optimization landscape. In particular, we present a variation of the Eckart-Young Theorem characterizing all critical points and their Hessian signatures for teacher-student problems with quadratic networks and Gaussian training data.

cs.LG↗

arXiv2024

Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor

The block tensor of trifocal tensors provides crucial geometric information on the three-view geometry of a scene. The underlying synchronization problem seeks to recover camera poses (locations and orientations up to a global transformation) from the block trifocal tensor. We establish an explicit Tucker factorization of this tensor, revealing a low multilinear rank of $(6,4,4)$ independent of the number of cameras under appropriate scaling conditions. We prove that this rank constraint provides sufficient information for camera recovery in the noiseless case. The constraint motivates a synchronization algorithm based on the higher-order singular value decomposition of the block trifocal tensor. Experimental comparisons with state-of-the-art global synchronization methods on real datasets demonstrate the potential of this algorithm for significantly improving location estimation accuracy. Overall this work suggests that higher-order interactions in synchronization problems can be exploited to improve performance, beyond the usual pairwise-based approaches.

cs.CV↗

arXiv2024

Fast expansion into harmonics on the ball

We devise fast and provably accurate algorithms to transform between an $N\times N \times N$ Cartesian voxel representation of a three-dimensional function and its expansion into the {ball harmonics}, that is, the eigenbasis of the Dirichlet Laplacian on the unit ball in $\mathbb{R}^3$. Given $\varepsilon > 0$, our algorithms achieve relative $\ell^1$ - $\ell^\infty$ accuracy $\varepsilon$ in time $O(N^3 (\log N)^2 + N^3 |\log \varepsilon|^2)$, while the na\"{i}ve direct application of the expansion operators has time complexity $O(N^6)$. We illustrate our methods on numerical examples.

math.NA↗

arXiv2024

Algebraic Constraints and Algorithms for Common Lines in Cryo-EM

We revisit the topic of common lines between projection images in single particle cryo-electron microscopy (cryo-EM). We derive a novel low-rank constraint on a certain $2n \times n$ matrix storing properly-scaled basis vectors for the common lines between $n$ projection images of one molecular conformation. Using this algebraic constraint and others, we give optimization algorithms to denoise common lines and recover the unknown 3D rotations associated to the images. As an application, we develop a clustering algorithm to partition a set of noisy images into homogeneous communities using common lines, in the case of discrete heterogeneity in cryo-EM. We demonstrate the methods on synthetic and experimental datasets.

math.OC↗

arXiv2024

Moment-based metrics for molecules computable from cryo-EM images

Single particle cryogenic electron microscopy (cryo-EM) is an imaging technique capable of recovering the high-resolution 3-D structure of biological macromolecules from many noisy and randomly oriented projection images. One notable approach to 3-D reconstruction, known as Kam's method, relies on the moments of the 2-D images. Inspired by Kam's method, we introduce a rotationally invariant metric between two molecular structures, which does not require 3-D alignment. Further, we introduce a metric between a stack of projection images and a molecular structure, which is invariant to rotations and reflections and does not require performing 3-D reconstruction. Additionally, the latter metric does not assume a uniform distribution of viewing angles. We demonstrate uses of the new metrics on synthetic and experimental datasets, highlighting their ability to measure structural similarity.

q-bio.BM↗

arXiv2023

Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications

Covering numbers are a powerful tool used in the development of approximation algorithms, randomized dimension reduction methods, smoothed complexity analysis, and others. In this paper we prove upper bounds on the covering number of numerous sets in Euclidean space, namely real algebraic varieties, images of polynomial maps and semialgebraic sets in terms of the number of variables and degrees of the polynomials involved. The bounds remarkably improve the best known general bound by Yomdin-Comte, and our proof is much more straightforward. In particular, our result gives new bounds on the volume of the tubular neighborhood of the image of a polynomial map and a semialgebraic set, where results for varieties by Lotz and Basu-Lerario are not directly applicable. We illustrate the power of the result on three computational applications. Firstly, we derive a near-optimal bound on the covering number of tensors with low canonical polyadic (CP) rank, quantifying their approximation properties and filling in an important missing piece of theory for tensor dimension reduction and reconstruction. Secondly, we prove a bound on dimensionality reduction of images of polynomial maps via randomized sketching, which has direct applications to large scale polynomial optimization. Finally, we deduce generalization error bounds for deep neural networks with rational or ReLU activation functions, improving or matching the best known results in the machine learning literature while helping to quantify the impact of architecture choice on generalization error.

math.AG↗