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Edited by William Bialek, Princeton University, Princeton, NJ, and approved May 31, 2019 (received for review October 5, 2018)
Significance
We introduce Intensive Principal Component Analysis (InPCA), a widely applicable manifold-learning method to visualize general probabilistic models and data. Using replicas to tune dimensionality in high-dimensional data, we use the zero-replica limit to discover a distance metric, which preserves distinguishability in high dimensions, and an embedding with superior visualization performance. We apply InPCA to the model of cosmology which predicts the angular power spectrum of the cosmic microwave background, allowing visualization of the space of model predictions (i.e., different universes).
Abstract
Unsupervised learning makes manifest the underlying structure of data without curated training and specific problem definitions. However, the inference of relationships between data points is frustrated by the “curse of dimensionality” in high dimensions. Inspired by replica theory from statistical mechanics, we consider replicas of the system to tune the dimensionality and take the limit as the number of replicas goes to zero. The result is intensive embedding, which not only is isometric (preserving local distances) but also allows global structure to be more transparently visualized. We develop the Intensive Principal Component Analysis (InPCA) and demonstrate clear improvements in visualizations of the Ising model of magnetic spins, a neural network, and the dark energy cold dark matter (
Footnotes
- ↵1To whom correspondence may be addressed. Email: knq2{at}cornell.edu.
Author contributions: K.N.Q., C.B.C., F.D.B., M.D.N., and J.P.S. designed research; K.N.Q. performed research; K.N.Q. contributed new reagents/analytic tools; K.N.Q. analyzed data; and K.N.Q., C.B.C., F.D.B., M.D.N., and J.P.S. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
Data deposition: All code used to generate the figures is available through public repositories on Github. Fig. 1 and Fig. 2 can be generated from code on https://github.com/katnquinn/Ising_ModelManifold, Fig. 3 can be generated from code on https://github.com/katnquinn/IntensiveEmbedding, Fig. 4 from code found on https://github.com/katnquinn/1Spin, and Fig. 5 from code on https://github.com/katnquinn/CMB_ModelManifold and from Code for Anisotropies in the Microwave Background software found at https://camb.info/.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1817218116/-/DCSupplemental.
Published under the PNAS license.
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Visualizing probabilistic models and data with Intensive Principal Component Analysis
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