Dimensionality Collapse: Optimal Measurement Selection for Low-Error Infinite-Horizon Forecasting

• URL: http://arxiv.org/abs/2303.15407v1
• Date: Mon, 27 Mar 2023 17:25:04 GMT
• Title: Dimensionality Collapse: Optimal Measurement Selection for Low-Error Infinite-Horizon Forecasting
• Authors: Helmuth Naumer and Farzad Kamalabadi
• Abstract summary: We solve the problem of sequential linear measurement design as an infinite-horizon problem with the time-averaged trace of the Cram'er-Rao lower bound (CRLB) for forecasting as the cost. By introducing theoretical results regarding measurements under additive noise from natural exponential families, we construct an equivalent problem from which a local dimensionality reduction can be derived. This alternative formulation is based on the future collapse of dimensionality inherent in the limiting behavior of many differential equations and can be directly observed in the low-rank structure of the CRLB for forecasting.
• Score: 3.5788754401889022
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This work introduces a method to select linear functional measurements of a vector-valued time series optimized for forecasting distant time-horizons. By formulating and solving the problem of sequential linear measurement design as an infinite-horizon problem with the time-averaged trace of the Cram\'{e}r-Rao lower bound (CRLB) for forecasting as the cost, the most informative data can be collected irrespective of the eventual forecasting algorithm. By introducing theoretical results regarding measurements under additive noise from natural exponential families, we construct an equivalent problem from which a local dimensionality reduction can be derived. This alternative formulation is based on the future collapse of dimensionality inherent in the limiting behavior of many differential equations and can be directly observed in the low-rank structure of the CRLB for forecasting. Implementations of both an approximate dynamic programming formulation and the proposed alternative are illustrated using an extended Kalman filter for state estimation, with results on simulated systems with limit cycles and chaotic behavior demonstrating a linear improvement in the CRLB as a function of the number of collapsing dimensions of the system.

Related papers

• Dimension reduction via score ratio matching [0.9012198585960441]
  We propose a framework, derived from score-matching, to extend gradient-based dimension reduction to problems where gradients are unavailable. We show that our approach outperforms standard score-matching for problems with low-dimensional structure.
  arXiv  Detail & Related papers  (2024-10-25T22:21:03Z)
• Learning Semilinear Neural Operators : A Unified Recursive Framework For Prediction And Data Assimilation [21.206744437644982]
  We propose a learning-based state-space approach to compute solution operators to infinite-dimensional semilinear PDEs. We develop a flexible method that allows for both prediction and data assimilation by combining prediction and correction operations. We show through experiments on Kuramoto-Sivashinsky, Navier-Stokes and Korteweg-de Vries equations that the proposed model is robust to noise and can leverage arbitrary amounts of measurements to correct its prediction over a long time horizon with little computational overhead.
  arXiv  Detail & Related papers  (2024-02-24T00:10:51Z)
• Large-Scale OD Matrix Estimation with A Deep Learning Method [70.78575952309023]
  The proposed method integrates deep learning and numerical optimization algorithms to infer matrix structure and guide numerical optimization. We conducted tests to demonstrate the good generalization performance of our method on a large-scale synthetic dataset.
  arXiv  Detail & Related papers  (2023-10-09T14:30:06Z)
• Low-rank extended Kalman filtering for online learning of neural networks from streaming data [71.97861600347959]
  We propose an efficient online approximate Bayesian inference algorithm for estimating the parameters of a nonlinear function from a potentially non-stationary data stream. The method is based on the extended Kalman filter (EKF), but uses a novel low-rank plus diagonal decomposition of the posterior matrix. In contrast to methods based on variational inference, our method is fully deterministic, and does not require step-size tuning.
  arXiv  Detail & Related papers  (2023-05-31T03:48:49Z)
• An evaluation framework for dimensionality reduction through sectional curvature [59.40521061783166]
  In this work, we aim to introduce the first highly non-supervised dimensionality reduction performance metric. To test its feasibility, this metric has been used to evaluate the performance of the most commonly used dimension reduction algorithms. A new parameterized problem instance generator has been constructed in the form of a function generator.
  arXiv  Detail & Related papers  (2023-03-17T11:59:33Z)
• Time varying regression with hidden linear dynamics [74.9914602730208]
  We revisit a model for time-varying linear regression that assumes the unknown parameters evolve according to a linear dynamical system. Counterintuitively, we show that when the underlying dynamics are stable the parameters of this model can be estimated from data by combining just two ordinary least squares estimates.
  arXiv  Detail & Related papers  (2021-12-29T23:37:06Z)
• Spike-and-Slab Generalized Additive Models and Scalable Algorithms for High-Dimensional Data [0.0]
  We propose hierarchical generalized additive models (GAMs) to accommodate high-dimensional data. We consider the smoothing penalty for proper shrinkage of curve and separation of smoothing function linear and nonlinear spaces. Two and deterministic algorithms, EM-Coordinate Descent and EM-Iterative Weighted Least Squares, are developed for different utilities.
  arXiv  Detail & Related papers  (2021-10-27T14:11:13Z)
• Unfolding Projection-free SDP Relaxation of Binary Graph Classifier via GDPA Linearization [59.87663954467815]
  Algorithm unfolding creates an interpretable and parsimonious neural network architecture by implementing each iteration of a model-based algorithm as a neural layer. In this paper, leveraging a recent linear algebraic theorem called Gershgorin disc perfect alignment (GDPA), we unroll a projection-free algorithm for semi-definite programming relaxation (SDR) of a binary graph. Experimental results show that our unrolled network outperformed pure model-based graph classifiers, and achieved comparable performance to pure data-driven networks but using far fewer parameters.
  arXiv  Detail & Related papers  (2021-09-10T07:01:15Z)
• Effective Dimension Adaptive Sketching Methods for Faster Regularized Least-Squares Optimization [56.05635751529922]
  We propose a new randomized algorithm for solving L2-regularized least-squares problems based on sketching. We consider two of the most popular random embeddings, namely, Gaussian embeddings and the Subsampled Randomized Hadamard Transform (SRHT)
  arXiv  Detail & Related papers  (2020-06-10T15:00:09Z)
• Bayesian System ID: Optimal management of parameter, model, and measurement uncertainty [0.0]
  We evaluate the robustness of a probabilistic formulation of system identification (ID) to sparse, noisy, and indirect data. We show that the log posterior has improved geometric properties compared with the objective function surfaces of traditional methods.
  arXiv  Detail & Related papers  (2020-03-04T22:48:30Z)
• Real-time Linear Operator Construction and State Estimation with the Kalman Filter [0.0]
  The proposed method uses three ideas: data in estimation an observation space, a time-invariant interval, and an online learning framework. By introducing localization and uniformity to the proposed method, we have demonstrated that noise can be reduced in high-dimensional spatial-temporal uniformity. The proposed method has potential for use in areas such as weather forecasting and vector field analysis.
  arXiv  Detail & Related papers  (2020-01-30T11:16:16Z)

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.