## An Alternative to EM for Gaussian Mixture Models

Victor Sanches Portella

MLRG @ UBC

August 11, 2021

## Why this paper?

Interesting application of Riemannian optimization to an important problem in ML

Details of the technicalities are not important

Yet, we have enough to sense the flavor of the field

## Gaussian Mixture Models

## Gaussian Mixture Models

A **GMM** is a convex combination of a fixed number of Gaussians

Image source: https://angusturner.github.io/assets/images/mixture.png

## Gaussian Mixture Models

A **GMM** is a convex combination of a fixed number of Gaussians

Density of a \(K\)-mixture on \(\mathbb{R}^d\):

Mean and Covariance of \(i\)-th Gaussian

Gaussian density

Weight of \(i\)-th Gaussian

(**Must** sum to 1)

**Find** GMM most likely to generate data points \(x_1, \dotsc, x_n \)

## Fitting a GMM

Image source: https://towardsdatascience.com/gaussian-mixture-models-d13a5e915c8e

**Find** GMM most likely to generate data points \(x_1, \dotsc, x_n \)

## Fitting a GMM

**Idea:** use means \(\mu_i\) and covariances \(\Sigma_i\) that maximize log-likelihood

**Hard to describe**

**Open - No clear projection**

Classical optimization methods (Newton, CG, IPM) struggle

## Optimization vs "\(\succ 0\)"

Slowdown when iterates get closer to boundary

IPM works, but it is slow

Force positive definiteness via "Cholesky decomposition"

Adds spurious maxima/minima

Slow

Expectation Maximization guarantees \(\succ 0\) "for free"

Easy for GMM's

Fast in practice

**EM**: Iterative method to find (local) max-likely parameters

## EM in one slide

**E-step**: Fix parameters, find "weighted log-likelihood"

**M-step**: Find parameters by maximizing the "weighted log-likelihood"

**For GMM's:**

**E-step**: Given \(\mu_i, \Sigma_i\), compute \(P(x_j \in \mathcal{N}_i)\)

**M-step**: Max weighted log-likelihood with weights \(P(x_j \in \mathcal{N}_i)\)

\(= p(x_j ; \mu_i, \Sigma_i)\)

CLOSED FORM SOLUTION FOR GMMs!

Guarantees \(\succ 0\) for free

## Riemannian Optimization

## Manifold of PD Matrices

## Manifold of PD Matrices

Rimennian metric at

Tangent space at a point

All symmetric matrices

Remark: Blows up when \(\Sigma\) is close to the boundary

## Optimization Structure

**General steps:**

1 - Find descent direction

2 - Perform line-search and step with retraction

Retraction: a way to move on a manifold along a tangent direction

Not necessarily follows a geodesic!

In this paper, we will only use the Exponential Map:

where

is the geodesic starting at \(\Sigma\) with direction \(D\)

## The Exponential Map

where

\(\gamma_D(t)\) is the geodesic starting at \(\Sigma\) with direction \(D\)

Geodesic between \(\Sigma\) and \(\Gamma\)

\(\Sigma\) at \(t = 0\) and \(\Gamma\) at \(t = 1\)

From this (?), we get the form of the exponential map

## Gradient Descent

GD on the Riemannian Manifold \(\mathbb{P}^d\):

Riemannian gradient of \(f\).

Depends on the metric!

If we have time, we shall see conditions for convergence of SGD

The methods the authors use are Conjugate Gradient and LBFGS.

Traditional gradient descent (GD):

Defining those depends on the idea of vector transport and Hessians. **I'll skip these descriptions**

## Riemannian opt for GMMs

We have a geometric structure on \(\mathbb{P}^d\). The other variable can stay in the traditional Euclidean space.

We can apply methods for Riemannian optimization!

... but they are way slower than traditional **EM**

## GMMs and Geodesic Convexity

## Single Gaussian Estimation

Maximum Likelihood estimation for a single Gaussian:

\(\mathcal{L}\) is concave!

In fact, the above has a **closed form solution **

In a Riemannian manifold, meaning of convexity changes

Geodesic convexity (g-convexity):

Geodesic from \(\Sigma_1\) to \(\Sigma_2\)

### \(\mathcal{L}\) is not g-concave

## Single Gaussian Estimation

Maximum Likelihood estimation for a single Gaussian:

\(\mathcal{L}\) is concave!

### \(\mathcal{L}\) is not g-concave

Solution: lift the function to a higher dimension!

### \(\hat{\mathcal{L}}\) is g-concave

## Single Gaussian Estimation

###
**Theorem** If \(\mu^*, \Sigma^*\) max \(\mathcal{L}\) and \(S^*\) maxs \(\hat{\mathcal{L}}\), then

### \(\mathcal{L}\) is not g-concave

### \(\hat{\mathcal{L}}\) is g-concave

### and

###
**Theorem** If \(\mu^*, \Sigma^*\) max \(\mathcal{L}\) and \(S^*\) maxs \(\hat{\mathcal{L}}\), then

### and

**Proof idea:**

Write

### Optimal choice of \(s\) is 1 **\(\implies\)** both \(\mathcal{L}\) and \(\hat{\mathcal{L}}\) agree

### PS: \(\Sigma^* \succ 0\) by Shur's complement Lemma

## Back to Many Gaussians

For now, **fix**

###
**Theorem** The local maxima of \(\mathcal{L}\) and of \(\hat{\mathcal{L}}\) agree

What about \(\alpha_1, \dotsc, \alpha_K\)? Reparameterize to use unconstrained opt

I think there are not many guarantees on final \(\alpha_j\)'s

## Plots!

### The effect of the lifting on convergence

## Plots!

### EM vs Riemannian Opt

PS: Reformulation helps Cholesky as well, but this information disappeared from the final version

## Remarks on What I Skipped

### Step-size Line Search

### Lifting with Regularization

### Details on vector transport and 2nd order methods

## Riemannian SGD

## Riemannian SGD

\(i_t\) a random index

Assumptions on \(f\):

Riemann Lip. Smooth

Unbiased Gradients

Bounded Variance

We want to minimize

**Riemannian Stochastic Gradient Descent:**

## Riemannian SGD

Assumptions on \(f\):

Riemann Lip. Smooth

Unbiased Gradients

Bounded Variance

**Riemannian SGD**

**Theorem **If one of the following holds:

**i) **Bounded gradients:

**ii)**** **Outputs an uniformly random iterate

Then

**\(\tau\) is unif. random if (ii)**

**\(\tau\) is best iter. if (i)**

(No need for bdd var)

## Does SGD work for GMM?

Do the conditions for SGD to converge hold for GMMs?

**Theorem **If \(\eta_t \leq 1\), then iterates stay in a compact set.

(For some reason they analyze SGD with a different retraction...)

**Remark **Objective function is not Riemannian Lip. smooth outside of a compact set

**Fact **Iterates stay in a compact set \(\implies\)

Bounded Gradients

\(\implies\)

Classical Lip. smoothness

\(\implies\)

Thm from Riemannian Geometry

Riemann Lip. smoothness

## It works!

With a different retraction and at a slower rate...?

#### MLRG - Riemannian opt

By Victor Sanches Portella

# MLRG - Riemannian opt

- 199