K-means method for Signal Compression: Vector Quantization.
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Transcript of K-means method for Signal Compression: Vector Quantization.
K-means method for Signal Compression: Vector Quantization
Blocks of signals: A sequence of audio. A block of image pixels.Formally: vector example: (0.2, 0.3, 0.5, 0.1)
A vector quantizer maps k-dimensional vectors in the vector space Rk into a finite set of vectors
Y = {yi: i = 1, 2, ..., N}.
Each vector yi is called a code vector or a codeword. and the set of all the codewords is called a codebook. Associated with each codeword, yi, is a nearest neighbor region called Voronoi region, and it is defined by:
The set of Voronoi regions partition the entire space Rk .
Voronoi Region
Codewords in 2-dimensional space. Input vectors are marked with an x, codewords are marked with red circles, and the Voronoi regions are separated with boundary lines.
Two Dimensional Voronoi Diagram
The Schematic of a Vector Quantizer (signal compression)
Compression Formula Amount of compression:
Codebook size is K, input vector of dimension L In order to inform the decoder of which code
vector is selected, we need to use bits. • E.g. need 8 bits to represent 256 code vectors.
Rate: each code vector contains the reconstruction value of L source output samples, the number of bits per vector component would be: .
K is called “level of vector quantizer”.
K2log
LK /log2
Vector Quantizer Algorithm
1. Determine the number of codewords, N, or the size of the codebook.
2. Select N codewords at random, and let that be the initial codebook. The initial codewords can be randomly chosen from the set of input vectors.
3. Using the Euclidean distance measure clusterize the vectors around each codeword. This is done by taking each input vector and finding the Euclidean distance between it and each codeword. The input vector belongs to the cluster of the codeword that yields the minimum distance.
Vector Quantizer Algorithm (contd.)
4. Compute the new set of codewords. This is done by obtaining the average of each cluster. Add the component of each vector and divide by the number of vectors in the cluster.
where i is the component of each vector (x, y, z, ... directions), m is the number of vectors in the cluster.
5. Repeat steps 2 and 3 until the either the codewords don't change or the change in the codewords is small.
Other Algorithms Problem: k-means is a greedy algorithm, may fall
into Local minimum. Four methods selecting initial vectors:
Random Splitting (with perturbation vector) Animation Train with different subset PNN (pairwise nearest neighbor)
Empty cell problem: No input corresponds to am output vector Solution: give to other clusters, e.g. most populate
cluster.
VQ for image compression
Taking blocks of images as vector L=NM. If K vectors in code book:
need to use bits. Rate:
The higher the value K, the better quality, but lower compression ratio.
Overhead to transmit code book:
Train with a set of images.
K2log
LK /log2
codebook size K Overhead bits/pixel16 0.0312564 0.125
256 0.51024 2
K-Nearest Neighbor Learning
22c:145University of Iowa
Different Learning Methods
Parametric Learning The target function is described by a set of
parameters (examples are forgotten) E.g., structure and weights of a neural network
Instance-based Learning Learning=storing all training instances Classification=assigning target function to a new
instance Referred to as “Lazy” learning
Instance-based Learning
Its very similar to aDesktop!!
General Idea of Instance-based Learning
Learning: store all the data instances Performance:
when a new query instance is encountered• retrieve a similar set of related instances
from memory• use to classify the new query
Pros and Cons of Instance Based Learning
Pros Can construct a different approximation to the
target function for each distinct query instance to be classified
Can use more complex, symbolic representations
Cons Cost of classification can be high Uses all attributes (do not learn which are
most important)
Instance-based Learning K-Nearest Neighbor Algorithm Weighted Regression Case-based reasoning
Most basic type of instance learning Assumes all instances are points in n-
dimensional space A distance measure is needed to
determine the “closeness” of instances Classify an instance by finding its
nearest neighbors and picking the most popular class among the neighbors
k-nearest neighbor (knn) learning
1-Nearest Neighbor
3-Nearest Neighbor
Important Decisions Distance measure Value of k (usually odd) Voting mechanism Memory indexing
Euclidean Distance
Typically used for real valued attributes Instance x (often called a feature
vector)
Distance between two instances xi and xj
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Discrete Valued Target Function
Training algorithm: For each training example <x, f(x)>, add the example to the list training_examples
Classification algorithm: Given a query instance xq to be classified. Let x1…xk be the k training examples nearest to xq
Return
otherwise 0),(
if 1),( where
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Continuous valued target function
Algorithm computes the mean value of the k nearest training examples rather than the most common value
Replace fine line in previous algorithm with
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Training dataset
Customer ID Debt Income Marital Status Risk
Abel High High Married Good
Ben Low High Married Doubtful
Candy Medium Very low Unmarried Poor
Dale Very high Low Married Poor
Ellen High Low Married Poor
Fred High Very low Married Poor
George Low High Unmarried Doubtful
Harry Low Medium Married Doubtful
Igor Very Low Very High Married Good
Jack Very High Medium Married Poor
k-nn K = 3 Distance
Score for an attribute is 1 for a match and 0 otherwise
Distance is sum of scores for each attribute
Voting scheme: proportionate voting in case of ties
Customer ID Debt Income Marital Status Risk
Abel High High Married Good
Ben Low High Married Doubtful
Candy Medium Very low Unmarried Poor
Dale Very high Low Married Poor
Ellen High Low Married Poor
Fred High Very low Married Poor
George Low High Unmarried Doubtful
Harry Low Medium Married Doubtful
Igor Very Low Very High Married Good
Jack Very High Medium Married Poor
Zeb High Medium Married ?
Query:
Customer ID Debt Income Marital Status Risk
Abel High High Married Good
Ben Low High Married Doubtful
Candy Medium Very low Unmarried Poor
Dale Very high Low Married Poor
Ellen High Low Married Poor
Fred High Very low Married Poor
George Low High Unmarried Doubtful
Harry Low Medium Married Doubtful
Igor Very Low Very High Married Good
Jack Very High Medium Married Poor
Yong Low High Married ?
Query:
Customer ID Debt Income Marital Status Risk
Abel High High Married Good
Ben Low High Married Doubtful
Candy Medium Very low Unmarried Poor
Dale Very high Low Married Poor
Ellen High Low Married Poor
Fred High Very low Married Poor
George Low High Unmarried Doubtful
Harry Low Medium Married Doubtful
Igor Very Low Very High Married Good
Jack Very High Medium Married Poor
Vasco High Low Married ?
Query:
Voronoi Diagram Decision surface formed by the training
examples of two attributes
Examples of one attribute
Distance-Weighted Nearest Neighbor Algorithm
Assign weights to the neighbors based on their ‘distance’ from the query point Weight ‘may’ be inverse square of the
distancesAll training points may influence a
particular instance Shepard’s method
Kernel function for Distance-Weighted Nearest Neighbor
Examples of one attribute
Remarks
+Highly effective inductive inference method for noisy training data and complex target functions
+Target function for a whole space may be described as a combination of less complex local approximations
+Learning is very simple- Classification is time consuming
Curse of Dimensionality
- When the dimensionality increases, the volume of the space increases so fast that the available data becomes sparse. This sparsity is problematic for any method that requires statistical significance.
Curse of Dimensionality
Suppose there are N data points of dimension n in the space [-1/2, 1/2]n.
The k-neighborhood of a point is defined to be the smallest hypercube containing the k-nearest neighbor.
Let l be the average side length of a k-neighborhood. Then the volume of an average hypercube is dn.
So dn/1n = k/N, or d = (k/N)1/n
d = (k/N)1/n
N k n d
1,000,000 10 2 0.003
1,000,000 10 3 0.02
1,000,000 10 17 0.5
1,000,000 10 200 0.94
When n is big, all the points are outliers.
- Curse of Dimensionality
- Curse of Dimensionality