Robust Automatic Differentiation of Square-Root Kalman Filters via Gramian Differentials

• URL: http://arxiv.org/abs/2603.13559v1
• Date: Fri, 13 Mar 2026 19:55:08 GMT
• Title: Robust Automatic Differentiation of Square-Root Kalman Filters via Gramian Differentials
• Abstract summary: Square-root Kalman filters propagate state covariances in Cholesky-factor form for numerical stability.<n>Their core operation, triangularization of a matrix $M in mathbbRn times m$, is computed via a QR decomposition in practice.<n>We extend it to rank-deficient inputs via a two-component decomposition: a column-space term based on the Moore--Penrose pseudoinverse, and a null-space correction for perturbations outside the column space of $M$.
• Score: 3.542663650865005
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Square-root Kalman filters propagate state covariances in Cholesky-factor form for numerical stability, and are a natural target for gradient-based parameter learning in state-space models. Their core operation, triangularization of a matrix $M \in \mathbb{R}^{n \times m}$, is computed via a QR decomposition in practice, but naively differentiating through it causes two problems: the semi-orthogonal factor is non-unique when $m > n$, yielding undefined gradients; and the standard Jacobian formula involves inverses, which diverges when $M$ is rank-deficient. Both are resolved by the observation that all filter outputs relevant to learning depend on the input matrix only through the Gramian $MM^\top$, so the composite loss is smooth in $M$ even where the triangularization is not. We derive a closed-form chain-rule directly from the differential of this Gramian identity, prove it exact for the Kalman log-marginal likelihood and filtered moments, and extend it to rank-deficient inputs via a two-component decomposition: a column-space term based on the Moore--Penrose pseudoinverse, and a null-space correction for perturbations outside the column space of $M$.

Related papers

• Orthogonal Polynomial Approximation for Matrix Log Normalization in Global Covariance Pooling [11.561251776738324]
  Global Covariance Pooling (GCP) improves deep networks by capturing second-order feature statistics, and is especially effective for fine-grained recognition.<n>Key ingredient is a mean-eigenvalue pre-normalization that centres the spectrum near 1, away from the singularity of log, with a post-compensation that returns the singular part of log(A) in closed form.
  arXiv  Detail & Related papers  (2026-08-19T15:17:46Z)
• A Provable Oracle-Free Quantum Algorithm for Nonlinear Dynamics on Hybrid Oscillator-Qubit Processors [14.814423713691623]
  We develop a hybrid qubit-qumode algorithm for nonlinear ordinary differential equations of the form $dotmathbfx=mathbff(mathbfx)$ with warped degree$L$.<n>The algorithm propagates the state density and returns the trajectory as the peak of that density in the small-noise limit.
  arXiv  Detail & Related papers  (2026-07-30T17:10:47Z)
• GPTQ-2D: Cubic-Time Two-Sided Adaptive Rounding [54.10836630612739]
  GPTQ, or equivalently Babai's nearest plane algorithm, rounds a real matrix to integers under a quadratic metric.<n>We present GPTQ-2D, which produces the identical rounded matrix in cubic time.
  arXiv  Detail & Related papers  (2026-07-29T15:34:00Z)
• Recovery of latent inner products from an anisotropic Gaussian random geometric graph [10.573657119155916]
  We study the problem of recovering latent inner products from a random geometric graph with anisotropic Gaussian latent points.<n>To address the undesired degree fluctuations amplified by the anisotropy of the latent points, we consider the doubly centered adjacency matrix of the graph.
  arXiv  Detail & Related papers  (2026-07-26T15:39:14Z)
• Near-Optimal Mode Scaling for Finite-Dimensional Boson Sampling via Lie-Algebraic Leakage Bounds [0.8594140167290097]
  Boson sampling on matter-based platforms requires encoding the bosonic modes in finite-dimensional local Hilbert spaces.<n>We develop a unified framework for non-interacting sampling on the irreducible representations of compact Lie groups.<n>We prove, in a Gaussian model of the transition matrix, that its spectral norm concentrates at $tildeO(stn)$ rather than the $O(n)$ worst-case of prior spin-based emulations.
  arXiv  Detail & Related papers  (2026-07-13T15:36:29Z)
• The Reverse Telescoping Coordinate System for Positive Definite Matrices: Geometry, Computation, and Generative Modeling [3.451964963586459]
  We design a new unconstrained coordinate system where a $ptimes p$ symmetric positive definite (SPD) matrix $$ is represented by a map $(x)=rmRT(x)$, with $x=(v,d,r)inmathbbRtimesmathbbR(p-1)timesmathbbRp(p-1)/2$.<n>This construction results in important properties not available in other charts, e.g., matrix logarithm, such as Jacobian depending on only the log-determin
  arXiv  Detail & Related papers  (2026-06-13T19:18:45Z)
• Efficient Mean Curvature Computation on High-Dimensional Data Manifolds [52.452902154360565]
  Estimating local mean curvature at each point of a high-dimensional dataset is a key ingredient of machine learning algorithms.<n>This paper introduces two complementary contributions that together reduce this cost by several orders of magnitude.<n> Experiments on real-world datasets confirm speedups of 50 to 300 times relative to the original implementation.
  arXiv  Detail & Related papers  (2026-06-04T16:04:31Z)
• Regularized Online RLHF with Generalized Bilinear Preferences [68.44113000390544]
  We consider the problem of contextual online RLHF with general preferences.<n>We adopt the Generalized Bilinear Preference Model to capture preferences via low-rank, skew-symmetric matrices.<n>We prove that the dual gap of the greedy policy is bounded by the square of the estimation error.
  arXiv  Detail & Related papers  (2026-02-26T15:27:53Z)
• High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spaces [48.465738895704455]
  We construct path matrices for decomposition of Cartesian tensors up to rank $n=9$ with reduced and affordable complexity.<n>Our method avoids the RREF algorithm and maintains a fully analytical derivation of each ICT decomposition matrix.<n>We extend our result to the arbitrary tensor product and direct sum spaces, enabling free design between different spaces while keeping symmetry.
  arXiv  Detail & Related papers  (2024-12-24T08:25:38Z)
• Accurate and Scalable Stochastic Gaussian Process Regression via Learnable Coreset-based Variational Inference [8.077736581030264]
  We introduce a novel inductive variational inference method for Gaussian process ($mathcalGP$) regression, by deriving a posterior over a learnable set of coresets.<n>Unlike former free-form variational families for inference, our coreset-based variational $mathcalGP$ (CVGP) is defined in terms of the $mathcalGP$ prior and the (weighted) data likelihood.
  arXiv  Detail & Related papers  (2023-11-02T17:22:22Z)
• The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions [32.30527731746912]
  We propose a novel approximate filtering and smoothing method which propagates lowrank approximations of low-rank matrices. Our method reduces computational complexity from cubic (for the Kalman filter) to emphquadratic in the state-space size in the worst-case.
  arXiv  Detail & Related papers  (2023-06-13T13:50:31Z)
• Purity decay rate in random circuits with different configurations of gates [0.0]
  We study purity decay -- a measure for bipartite entanglement -- in a chain of $n$ under the action of various geometries. In most circuits, purity decays to its value in two stages: the initial thermodynamically relevant decay up to extensive times is $sim lambda_mathrmeffeff$, with $lambda_mathrmeff$ not necessarily being in the spectrum of the transfer matrix.
  arXiv  Detail & Related papers  (2022-11-24T12:32:07Z)
• Hybrid Model-based / Data-driven Graph Transform for Image Coding [54.31406300524195]
  We present a hybrid model-based / data-driven approach to encode an intra-prediction residual block. The first $K$ eigenvectors of a transform matrix are derived from a statistical model, e.g., the asymmetric discrete sine transform (ADST) for stability. Using WebP as a baseline image, experimental results show that our hybrid graph transform achieved better energy compaction than default discrete cosine transform (DCT) and better stability than KLT.
  arXiv  Detail & Related papers  (2022-03-02T15:36:44Z)
• High-Dimensional Gaussian Process Inference with Derivatives [90.8033626920884]
  We show that in the low-data regime $ND$, the Gram matrix can be decomposed in a manner that reduces the cost of inference to $mathcalO(N2D + (N2)3)$. We demonstrate this potential in a variety of tasks relevant for machine learning, such as optimization and Hamiltonian Monte Carlo with predictive gradients.
  arXiv  Detail & Related papers  (2021-02-15T13:24:41Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.