Homework #03: The K-means algorithm and Mixture of Gaussian
estimation with Expectation-Maximization algorithm.
The purpose of this assignment is to implement the K-means
algorithm and the Mixture of Gaussians estimation using
the Expectation-Maximization algorithm.
Your assignment should be sent by email to jhuangfu@cs.nyu.edu
- the DEADLINE IS APRIL 16
- include "G22-3033-014 homework 03" in the subject line
- Send your email in plain text (no msword, no html, no postscript, no pdf).
- late submissions will be penalized with the following formula:
corrected_grade = actual_grade / (1 + days_late/15).
- you must implement the code by yourself, but you are
encouraged to discuss your results with other students.
- Include your source code as attachment(s).
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PART 1: Implement K-means
The K-means algorithm is as follows.
Given a set of P training vectors X1...Xp
- initialize K prototype vectors P1...Pk
by setting each of them to a different
randomly chosen training sample
- repeat until convergence:
- for all i in [1..p], for all j in [1..k],
compute the responsabilities
Rij = 1 if Xi is closest to Pj than to any other prototype
Rij = 0 otherwise.
- recompute the prototypes so that each prototype is
equal to the mean of all the samples assigned to it
Pj = 1/[sum_i Rij] sum_i Rij.Xi
PART 2: Implement EM to estimate the parameters of a Mixture of Gaussians
Given a set of P training vectors X1...Xp
the likelihood of the data modeled by a MoG with k
component gaussians is equal to:
P(X1...Xp | W1...Wk,M1...Mk,A1...Ak) = product_i sum_j Wj.gaussian(Xi,Mj,Aj)
where i ranges over [1,p], and j over [1,k] (k is the
number of components in the mixture). gaussian(Xi,Mj,Aj) is
a multivariate Gaussian density with mean Mj and covariance matrix
Aj evaluated at point Xi. To ensure normalization, the Wj must be
between 0 and 1, and must verify: sum_j Wj = 1
The negative log-likelihood (energy) is:
L(W1...Wk,M1...Mk,A1...Ak) = - sum_i log[ sum_j Wj gaussian(Xi,Mj,Aj) ]
The EM algorithm for a MoG is as follows:
- initialize K prototype vectors P1...Pk
by running the K-means algorithm.
- repeat until convergence:
- for all i in [1..p], for all j in [1..k],
compute the responsabilities
Rij = Wj*gaussian(Xi,Mj,Aj) / sum_h Wh*gaussian(Xi,Mh,Ah)
- recompute the parameters of the gaussians:
- set Mj to the mean of all the samples *weighted* by their
responsabilities Rij
- set Aj to the covariance of all the samples *weighted* by
their responsabilities Rij.
- set Wj to [sum_i Rij]/[sum_j sum_i Rij]
MoG trained with EM can be seen as a "soft" version of K-means
where the responsabilities are continuous numbers between 0 and
1 (that sum to one over j) instead of being binary.
PART 3: Image compression using vector quantization
A simple technique for image compression consists in
(1) cutting up an image into small non-overlapping tiles
(e.g. 8 by 8 pixels), (2) viewing this set of tiles as
a dataset of 64-dimensional vectors (each vector component
being a pixel) (3) running the K-means algorithm on this
dataset (4) coding each tile by a code that represents
the index of the nearest prototype to the tile.
3.1: the image testimage.pgm or testimage.bmp (800x600 pixels, grayscale)
was cut up into 7500 non-overlapping, 8x8-pixel tiles. Those tiles
were turned into 64-dimensional vectors (by lining up the pixels in
raster order: left to right and top to bottom), and put together into
the Lush .mat file tiles.mat. The tiles are in raster order (left
to right, top to bottom), so reconstructing the image consists
in painting those tiles next to each other on a 100 columns by
75 line grid.
Run the K-means algorithm with K=2, 4 and 8 on that set
(be careful how you initialize the prototypes).
For each size, report the mean-squared error incurred by
replacing each tile by its closest prototype, i.e. the
average of ||Xi - Tk(i)||^2 over i, where Xi is the i-th tile,
and Tk(i) is the prototype (among the K prototypes) that
is closest to Xi.
3.2: for K-8, compute the binary entropy of the resulting
sequence of code for the whole image: make a integer
vector Zi with as many elements as samples in the dataset, where
each component contains the index of the prototype closest
to the corresponding tile [ k(i) ]. Then compute
the normalized histogram of the values in that vector
(i.e. a vector Hk where element k is the proportion
of elements in Zi that are equal to k (k in [0,7]).
Compute the binary entropy of the histogram:
Entropy = -sum_k Hk*log2[Hk]
where log2[x] is the base 2 logarithm of x.
report this value. This is the number of bits we would need
to compress the image if we had a perfect entropy coder
to turn those integers into short bit strings
(a perfect huffman coder or arithmetic coder).
PART 4: Testing EM for a Mixture of Gaussians.
Starting from the result obtained for question 3.2
(result of K-means with K=8 on the tile dataset)
train a mixture of Gaussians with 8 components
using EM on the tile dataset.
EM finds the parameters that minimize the negative
log-likelihood of the dataset given the model:
L = - sum_i log( ModelProb(Xi) )
Report the initial value of the negative log-likelihood
(after running K-means and one M step to compute the
covariance matrices, using the binary responsabilities
produced by K-means).
and the value at the end of the convergence of EM.
NOTE 1: computing a multivariate Gaussian involves
computing the inverse and the determinant of the covariance
matrix. This can be done in Lush with:
(libload "libnum/linalgebra")
(inverse-lu (idx-copy a) a-inverse)
(determinant (idx-copy a))
The copy is necessary because inverse-lu and determiant
destroy the input matrix.
NOTE 2: In some cases, the covariance matrix can be singular
(e.g. if the number of samples attached to a prototype is too small).
In that case, the covariance matrix can be "regularized" by adding
a small positive constant to the diagonal:
regularized_covar = origianl_covar + epsilon*identity_matrix
NOTE 3: in 64 dimension, the Gaussian likelihood of a vector
drops very, very fast with the distance. Numerical
problems may ensue. It might be advisable to do most computations
in log space (hint: Lush function determinant-log computes the log
determinant of a matrix directly).
PART 5: (extra credit): reconstruct the "compressed"
test image obtained by replacing each tile by its
closest prototype. Save the image in a pgm file.
Saving a PGM image file can be done in Lush with
(pgm-write-ubim "image.pgm" image)
where image is a 600x800 ubyte-matrix.
Attach the reconstructed image to your homework email.
NOTE: grayscale and RGB images can be displayed in a
Lush graphic window with
(new-window 0 0 800 600 "stuff")
(rgb-draw-matrix 0 0 image)