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<!DOCTYPE html>
<html>
<head>
<title>LOL</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/>
<script src="remark-latest.min.js" type="text/javascript"></script>
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<textarea id="source">
class: center, middle
name:opening
### Supervised Manifold Learning Outperforms PCA for Subsequent Inference
Joshua T. Vogelstein*, Minh Tang, Da Zheng, Randal Burns, Mauro Maggioni
<br>
.center[
<br>
<!-- JHU Kavli Neuroscience Discovery Institute -->
{[bme](http://www.bme.jhu.edu/), [icm](http://icm.jhu.edu/), [cis](http://cis.jhu.edu/), [idies](http://idies.jhu.edu/), [kavli](http://kavlijhu.org/), [cs](http://engineering.jhu.edu/computer-science/), [ams](http://engineering.jhu.edu/ams/), [neuro](http://neuroscience.jhu.edu/)} | [jhu](https://www.jhu.edu/)
<br>
questions: [[email protected]](mailto:jovo at jhu dot edu)
<br>
slides: <http://neurodata.io/tools/LOL/>
<br>
Co-Founder: [NeuroData](http://neurodata.io) & [gigantum](http://gigantum.io)
]
---
class: center, middle
# Interrupt!
---
### What is supervised learning?
Given (X<sub>i</sub>,Y<sub>i</sub>) pairs with neither F<sub>Y</sub> nor F<sub>Y</sub> degenerate,
*supervised learning* is the estimation of any given functional of F<sub>X|Y</sub>
--
#### Examples
- Classification
- Regression
- 2-sample testing
- K-sample testing
---
### Classification and Fisher's LDA
- Given F<sub>X|Y</sub> = N(μ<sub>y</sub>,Σ) and F<sub>Y</sub> = B(π),
- where X ϵ R<sup>p</sup>
- Bayes optimal classifier is x' Σ<sup>-1</sup> δ > t, where δ=μ<sub>0</sub>-μ<sub>1</sub>
--
#### Properties
- simple
- multiclass generalizations
- plug-in estimate converges to Bayes optimal
- algorithmic efficiency
---
### But...
- When n < p, our estimate of Σ is singular
- Cannot use Fisher's LDA
- What to do?
<br>
--
- Manifold learning
- Spare modeling (is secretly also manifold learning)
<br>
---
### Manifold Learning for Subsequent Inference
<img src="../Figs/mnist2.png" STYLE="margin:auto; width:100%"/>
---
### Limitations of existing approaches
- Manifold learing
- is typically unsupervised
- out of sample embedding is icky
- do not scale to terabytes (often require n<sup>3</sup> operations)
- who says directions of variance are near directions of discrimination?
<br>
--
- Sparse modeling (is supervised, but...)
- NP-hard (feature screening), or
- approximations do not scale to terabytes (Lasso), or
- non-convex (Dictionary learning),
- with icky hyperparameters (elastic net & dictionary learning)
---
## Linear Projections
- PCA: eig({x<sub>i</sub> - μ})
- PCA': eig({x<sub>i</sub> - μ<sub>j</sub>})
- LOL: [δ, eig({x<sub>i</sub> - μ<sub>j</sub>})]
"Linear Optimal Low-Rank"
<br>
--
### Notes
- Fisher's LDA uses δ and {x<sub>i</sub> - μ<sub>j</sub>}
- PCA' removes δ
- PCA kind of accidentally includes δ, Σ + π(1-π) δ δ', but weights it suboptimally
- LOL uses both terms explicitly, weighting δ more
- For each we compose with LDA on low-d estimates
---
## LOL Gaussian Intuition
<img src="../Figs/cigars_est.png" STYLE="margin:auto; width:90%"/>
---
## LOL > PCA Theory
<img src="LOL_theory.png" STYLE="margin:auto; width:100%"/>
---
## Unpacking the theory: Chernoff Information
- C(F,G) = sup<sub> t </sub> [ -log ∫ f<sup>t</sup>(x) g<sup>1-t</sup>(x) dx], for 0 < t < 1
- the *exponential rate* at which the Bayes error decreases
- it is the tightest possible bound on performance
- if F=N(μ<sub>0</sub>,Σ<sub>0</sub>) and G=N(μ<sub>1</sub>,Σ<sub>1</sub>)
- C(F,G)= 0.5 sup<sub>t</sub> t(1-t) δ' Σ<sup>-1</sup> δ + log |Σ<sub>t</sub>| / (|Σ<sub>0</sub>|<sup>t</sup> |Σ<sub>1</sub>|<sup>1-t</sup>)
--
### Chernoff on Projected Data
<!-- - let F =N(μ<sub>0</sub>,Σ) and G=N(μ<sub>1</sub>,Σ) -->
- let F & G be Gaussian with same covariance (LDA model)
- let A be any linear transformation
- C(F<sup>A</sup>, G<sup>A</sup>) = 1/8 * || P<sub>Z</sub> Σ<sup>-1/2</sup> δ ||<sub>F</sub><sup>2</sup>
- P<sub>Z</sub> = Z (Z' Z)<sup>-1</sup> Z', and Z = Σ<sup>1/2</sup>A'
- let C<sup>A</sup> := C(F<sup>A</sup>, G<sup>A</sup>)
---
## "Thm A: LOL > PCA'"
<!-- - let A = LOL projection -->
<!-- - let B = PCA' projection -->
- let F & G be Gaussian with same covariance
- C<sup>LOL</sup> ≥ C<sup>PCA'</sup>
- inequality is strict whenever δ' (I - U<sub>d</sub>U<sub>d</sub>' ) δ ≥ 0.
---
## "Thm B: LOL > PCA"
<!-- - let C = PCA projection -->
<!-- - let F & G be Gaussian with same covariance -->
- C<sup>PCA</sup> = 4K / (4 - K), where
- K = δ' Σ<sup>t</sup><sub>d</sub> δ
- so when δ is in the space spanned by the smaller eigenvectors, PCA discards nearly all the info
<br>
--
### Simple Example
- if Σ is diagonal with decreasing λ's,
- δ=(0,...,0,s),
- λ<sub>p</sub> + s<sup>2</sup>/4 < λ<sub>d</sub>
- C<sup>PCA</sup> = 0
- C<sup>LOL</sup> = s<sup>2</sup> / λ<sub>p</sub>
---
## "Thm C: [δ, U<sub>d</sub>] > U<sub>d</sub>"
- let U<sub>d</sub> be any matrix in R<sup>p x d</sup> with U<sub>d</sub> U<sub>d</sub>' = I
- arthimetic is messier
- nearly the same result as PCA
- basically, when δ and U<sub>d</sub> are nearly orthogonal, adding delta helps
---
### "Thm D: LOL > PCA as p increases"
- let γ = λ<sub>d</sub> - λ<sub>d+1</sub>
- let δ be sparse with probabilty ε an element is 0,
- o.w. it is Gaussian
<!-- with mean τ and standard deviation σ -->
- let p(1-ε) → θ
- then with probability at least ε<sup>d</sup>, C<sup>PCA</sup> = 0 < C<sup>LOL</sup>
- and this probability can be made arbitrarily close to 1
---
## "Thm E: LOL > PCA as n & p increases"
- when n/p → 0, all results trivially hold
<br>
- Suppose Σ is low rank + σ<sup>2</sup>I
- Suppose that: M log p ≤ log n ≤ M' log λ,
- provided M & M' are large enough
- estimates of C converge, so
- E [ C<sup>LOL</sup> ] > E[ C<sup>PCA</sup>]
---
## LOL > PCA Simulations
<img src="../Figs/plot_sims.png" STYLE="margin:auto; width:380px"/>
---
## LOL is fast
<img src="../Figs/scalability.png" STYLE="margin:auto; width:100%"/>
- utilize FlashX semi-external memory (SEM) computing
- optimal (linear) scale up and out
- SEM speed ≈ internal memory speed
- swap random projections (RP) with SVD for 10x speed improvement
- RP error rate ≈ SVD error rate
- ~3 minutes for a 2 terabyte dataset
---
## LOL > PCA Data
<img src="../Figs/plot_real.png" STYLE="margin:auto; WIDTH:100%;"/>
---
### LOL Hypothesis Testing & Regression
<img src="../Figs/regression_power.png" STYLE="margin:auto; WIDTH:100%;"/>
---
## Discussion
- simple supervised inference for wide data
- big data tools (eg, Spark, H20, VW) typically focus on large n
- algorithmic and theoretical generalizations straightforward
- open source implementations
<br>
<center>
<a href="http://neurodata.io/tools/LOL/">http://neurodataio/tools/LOL/</a>
</center>
---
class: center
<br>
# Questions?
## Hiring Postdocs & Software Engineers Now!
e: [[email protected]](mailto:[email protected]) |
w: [neurodata.io](http://neurodata.io), [gigantum.io](http://gigantum.io)
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