Statistical mechanics of high dimensional inference
Abstract/Contents
- Abstract
- Remarkable advances in measurement technologies have thrust us squarely into the modern age of big-data, which has the potential to revolutionize a variety of fields spanning the sciences, engineering, and the humanities. However, the advent of large scale data sets presents severe statistical challenges which must be solved if we are to gain conceptual insights from such data. A fundamental origin of the difficulty in analyzing many large scale data sets lies in their high dimensionality: often we simultaneously measure many variables, or dimensions, but only under a limited number of experimental conditions, or repetitions. For example, in neuroscience, we can now measure thousands of neurons simultaneously, but only under a limited number of behavioral trials. Classical statistics provides well defined prescriptions for finding patterns in low dimensional data sets in which a small number of variables are measured under many conditions or repetitions, but it does not provide analogous optimal guidance for high dimensional data. We use techniques from the statistical mechanics of quenched disorder, particularly replica theory, to derive optimal algorithms for high dimensional inference. We focus on the case of regression and derive fundamental limits on the performance of high dimensional regression with prior information as well as the exact form of optimal algorithms that achieve these limits. Our optimal algorithms are applicable beyond Gaussian distributions of signal and noise, and they out-perform widely cherished algorithms like maximum likelihood (ML) inference and maximum a-posteriori (MAP). Moreover, our results show how central mathematical objects in convex optimization theory and random matrix theory emerge naturally from statistical physics.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2016 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Advani, Madhu |
|---|---|
| Associated with | Stanford University, Department of Applied Physics. |
| Primary advisor | Ganguli, Surya, 1977- |
| Thesis advisor | Ganguli, Surya, 1977- |
| Thesis advisor | Baccus, Stephen A |
| Thesis advisor | Fisher, Daniel |
| Advisor | Baccus, Stephen A |
| Advisor | Fisher, Daniel |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Madhu Advani. |
|---|---|
| Note | Submitted to the Department of Applied Physics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2016. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2016 by Madhu Advani
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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