Quantifying Behavioural Distance Between Mathematical Expressions

• URL: http://arxiv.org/abs/2408.11515v1
• Date: Wed, 21 Aug 2024 10:48:04 GMT
• Title: Quantifying Behavioural Distance Between Mathematical Expressions
• Abstract summary: This paper proposes and implements a measure of a behavioral distance, BED, that clusters together expressions with similar errors. Our findings also reveal that BED significantly improves the smoothness of the error landscape in the search space for symbolic regression.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Existing symbolic regression methods organize the space of candidate mathematical expressions primarily based on their syntactic, structural similarity. However, this approach overlooks crucial equivalences between expressions that arise from mathematical symmetries, such as commutativity, associativity, and distribution laws for arithmetic operations. Consequently, expressions with similar errors on a given data set are apart from each other in the search space. This leads to a rough error landscape in the search space that efficient local, gradient-based methods cannot explore. This paper proposes and implements a measure of a behavioral distance, BED, that clusters together expressions with similar errors. The experimental results show that the stochastic method for calculating BED achieves consistency with a modest number of sampled values for evaluating the expressions. This leads to computational efficiency comparable to the tree-based syntactic distance. Our findings also reveal that BED significantly improves the smoothness of the error landscape in the search space for symbolic regression.

Related papers

• Representation of syntax in LLMs through the lens of linear distance and similarity-aware entropy [4.794327378667608]
  Structural probes were introduced by Hewitt and Manning to reconstruct syntactic trees from a neural language model's latent representations.<n>They are evaluated by calculating the proportion of syntactic tree edges correctly reconstructed over an annotated corpus.<n>Here, we disaggregate this measure, considering undirected attachment score by label (UASL), which assesses the reconstruction accuracy of each syntactic relation separately.
  arXiv  Detail & Related papers  (2026-08-28T01:17:18Z)
• Robust Graph Matching through Semantic Relationship Generation for SLAM [0.15635627702544694]
  Scene Graphs enable localization in structured indoor environments by matching a locally observed graph to a prior map.<n>We propose a semantic-enhanced graph matching approach that explicitly models relations between detected objects and structural elements.<n>Results show that semantic relations significantly reduce the number of candidate matches, improve computational efficiency, and enable faster convergence.
  arXiv  Detail & Related papers  (2026-04-28T09:15:51Z)
• Implicit score matching meets denoising score matching: improved rates of convergence and log-density Hessian estimation [5.773269033551628]
  We study the problem of estimating the score function using both implicit score matching and denoising score matching.<n>We prove that implicit score matching is able not only to adapt to the intrinsic dimension, but also to achieve the same rates of convergence as denoising score matching.
  arXiv  Detail & Related papers  (2025-12-30T17:39:48Z)
• From STLS to Projection-based Dictionary Selection in Sparse Regression for System Identification [1.7341202786497238]
  We revisit dictionary-based sparse regression, in particular, Sequential Threshold Least Squares (STLS)<n>We propose a score-guided library selection to provide practical guidance for data-driven modeling, with emphasis on SINDy-type algorithms.
  arXiv  Detail & Related papers  (2025-12-16T13:42:10Z)
• Efficient Covariance Estimation for Sparsified Functional Data [51.69796254617083]
  proposed Random-knots (Random-knots-Spatial) and B-spline (Bspline-Spatial) estimators of the covariance function are computationally efficient.<n>Asymptotic pointwise of the covariance are obtained for sparsified individual trajectories under some regularity conditions.
  arXiv  Detail & Related papers  (2025-11-23T00:50:33Z)
• Measuring similarity between embedding spaces using induced neighborhood graphs [10.056989400384772]
  We propose a metric to evaluate the similarity between paired item representations. Our results show that accuracy in both analogy and zero-shot classification tasks correlates with the embedding similarity.
  arXiv  Detail & Related papers  (2024-11-13T15:22:33Z)
• The Inefficiency of Genetic Programming for Symbolic Regression -- Extended Version [0.0]
  We analyse the search behaviour of genetic programming for symbolic regression in practically relevant but limited settings. This enables us to quantify the success probability of finding the best possible expressions. We compare the search efficiency of genetic programming to random search in the space of semantically unique expressions.
  arXiv  Detail & Related papers  (2024-04-26T09:49:32Z)
• Sparse Variational Student-t Processes [8.46450148172407]
  Student-t Processes are used to model heavy-tailed distributions and datasets with outliers. We propose a sparse representation framework to allow Student-t Processes to be more flexible for real-world datasets. We evaluate two proposed approaches on various synthetic and real-world datasets from UCI and Kaggle.
  arXiv  Detail & Related papers  (2023-12-09T12:55:20Z)
• Learning Unnormalized Statistical Models via Compositional Optimization [73.30514599338407]
  Noise-contrastive estimation(NCE) has been proposed by formulating the objective as the logistic loss of the real data and the artificial noise. In this paper, we study it a direct approach for optimizing the negative log-likelihood of unnormalized models.
  arXiv  Detail & Related papers  (2023-06-13T01:18:16Z)
• Score-based Continuous-time Discrete Diffusion Models [102.65769839899315]
  We extend diffusion models to discrete variables by introducing a Markov jump process where the reverse process denoises via a continuous-time Markov chain. We show that an unbiased estimator can be obtained via simple matching the conditional marginal distributions. We demonstrate the effectiveness of the proposed method on a set of synthetic and real-world music and image benchmarks.
  arXiv  Detail & Related papers  (2022-11-30T05:33:29Z)
• Data-driven Numerical Invariant Synthesis with Automatic Generation of Attributes [0.0]
  We propose a data-driven algorithm for numerical invariant synthesis and verification. The algorithm is based on the ICE-DT schema for learning decision trees from samples of positive and negative states.
  arXiv  Detail & Related papers  (2022-05-30T09:18:37Z)
• Scalable Intervention Target Estimation in Linear Models [52.60799340056917]
  Current approaches to causal structure learning either work with known intervention targets or use hypothesis testing to discover the unknown intervention targets. This paper proposes a scalable and efficient algorithm that consistently identifies all intervention targets. The proposed algorithm can be used to also update a given observational Markov equivalence class into the interventional Markov equivalence class.
  arXiv  Detail & Related papers  (2021-11-15T03:16:56Z)
• Optimal oracle inequalities for solving projected fixed-point equations [53.31620399640334]
  We study methods that use a collection of random observations to compute approximate solutions by searching over a known low-dimensional subspace of the Hilbert space. We show how our results precisely characterize the error of a class of temporal difference learning methods for the policy evaluation problem with linear function approximation.
  arXiv  Detail & Related papers  (2020-12-09T20:19:32Z)
• A similarity-based Bayesian mixture-of-experts model [0.5156484100374058]
  We present a new non-parametric mixture-of-experts model for multivariate regression problems. Using a conditionally specified model, predictions for out-of-sample inputs are based on similarities to each observed data point. Posterior inference is performed on the parameters of the mixture as well as the distance metric.
  arXiv  Detail & Related papers  (2020-12-03T18:08:30Z)
• Asymptotic Analysis of an Ensemble of Randomly Projected Linear Discriminants [94.46276668068327]
  In [1], an ensemble of randomly projected linear discriminants is used to classify datasets. We develop a consistent estimator of the misclassification probability as an alternative to the computationally-costly cross-validation estimator. We also demonstrate the use of our estimator for tuning the projection dimension on both real and synthetic data.
  arXiv  Detail & Related papers  (2020-04-17T12:47:04Z)
• Spatially Adaptive Inference with Stochastic Feature Sampling and Interpolation [72.40827239394565]
  We propose to compute features only at sparsely sampled locations. We then densely reconstruct the feature map with an efficient procedure. The presented network is experimentally shown to save substantial computation while maintaining accuracy over a variety of computer vision tasks.
  arXiv  Detail & Related papers  (2020-03-19T15:36:31Z)

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.