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Visualizing High-Dimensional Functions with Dense Maps

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Abstract

Multivariate functions have a central place in the development of techniques present many domains, such as machine learning and optimization research. However, only a few visual techniques exist to help users understand such multivariate problems, especially in the case of functions that depend on complex algorithms and variable constraints. In this paper, we propose a technique that enables the visualization of high-dimensional surfaces defined by such multivariate functions using a two-dimensional pixel map. We demonstrate two variants of it, OptMap, focused on optimization problems, and RegSurf, focused on regression problems in machine learning. Both our techniques are simple to implement, computationally efficient, and generic with respect to the nature of the high-dimensional data they address. We show how the two techniques can be used to visually explore a wide variety of optimization and regression problems.

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Availability of Data and Materials

Not applicable.

Code Availability

Our implementation, plus all codes used in our experiments, are publicly available at github.com/mespadoto/optmap.

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Funding

This study was financed in part by FAPESP under Grant Nos. 2015/22308-2, 2017/25835-9, and 2020/13275-1, and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.

Author information

Author notes

  1. Mateus Espadoto, Francisco C. M. Rodrigues, Nina S. T. Hirata, and Alexandru C. Telea have contributed equally to this work.

Authors and Affiliations

  1. Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010, São Paulo, 05508-090, Brazil

    Mateus Espadoto, Francisco C. M. Rodrigues & Nina S. T. Hirata

  2. Department of Information and Computing Sciences, Utrecht University, Princetonplein 5, 3584 CC, Utrecht, The Netherlands

    Alexandru C. Telea

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  1. Mateus Espadoto
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  2. Francisco C. M. Rodrigues
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This article is part of the topical collection “Computer Vision, Imaging and Computer Graphics Theory and Applications” guest edited by Jose Braz, A. Augusto Sousa, Alexis Paljic, Christophe Hurter, and Giovanni Maria Farinella.

Appendix 1: Implementation details

Appendix 1: Implementation details

Table 8 Software used for the OptMap and RegSurf implementation
Full size table

We implemented OptMap and RegSurf in Julia  [69] using the open-source software libraries in Table 8. The optimization examples (“OptMap: Test via High‑Dimensional Functions”, OptMap: Solvers for Unconstrained Problems and “OptMap Performance”) were implemented using Optim  [70] for the unconstrained problems, and JuMP  [5] for the constrained problems, using the solvers Clp  [11], Cbc  [12], GLPK  [13], and Ipopt  [52]. The regression examples (“RegSurf: Real‑World Datasets” and “RegSurf: Visualizing Overfitting”) were implemented using the MLJ library  [71]. Our implementation, plus all code used in our experiments, are publicly available at github.com/mespadoto/optmap.

The scalability experiments discussed in “OptMap Performance” and “RegSurf Performance” were executed, respectively, on a 4-core Intel Xeon E3-1240 v6 at 3.7 GHz with 64 GB RAM, and on a dual 16-core Intel Xeon Silver 4216 at 2.1 GHz with 256 GB RAM.

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Espadoto, M., Rodrigues, F.C.M., Hirata, N.S.T. et al. Visualizing High-Dimensional Functions with Dense Maps. SN COMPUT. SCI. 4, 230 (2023). https://doi.org/10.1007/s42979-022-01664-2

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