Voronoi Region 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 R k into a finite set of vectors Y = {y i : i = 1, 2,..., N}. Each vector y i is called a code vector or a codeword. and the set of all the codewords is called a codebook. Associated with each codeword, y i, is a nearest neighbor region called Voronoi region, and it is defined by: The set of Voronoi regions partition the entire space R k. Two Dimensional Voronoi Diagram The Schematic of a Vector Quantizer (signal compression) 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. 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 log 2 K 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: log 2 K / L. K is called level of vector quantizer. 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. 1
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 log 2 K bits. Rate: log 2 K / L The higher the value K, the better quality, but lower compression ratio. Overhead to transmit code book: codebook size K Overhead bits/pixel 16 0.03125 64 0.125 256 0.5 1024 2 Train with a set of images. K-Nearest Neighbor Learning 22c:145 University 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 a Desktop!! 2
General Idea of Instancebased 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 k-nearest neighbor (knn) learning 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 1-Nearest Neighbor 3-Nearest Neighbor 3
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) a x), a ( x), a ( ) 1( 2 n x Distance between two instances x i and x j d n ( xi, x j ) ( ar ( xi ) ar ( x j r 1 )) 2 Discrete Valued Target Function Continuous 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 x q to be classified. Let x 1 x k be the k training examples nearest to x q Return k fˆ( x ) arg max ( v, f ( x )) q v V i 1 where ( a, b) 1if a b ( a, b) 0 otherwise i Algorithm computes the mean value of the k nearest training examples rather than the most common value Replace fine line in previous algorithm with k f ( xi ) i fˆ( xq ) 1 k Training dataset 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 4
Query: Zeb High Medium Married? Query: Yong Low High Married? Query: Vasco High Low Married? 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 distances All training points may influence a particular instance Shepard s method 5
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 d n. So d n /1 n = 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. 6
- Curse of Dimensionality - Curse of Dimensionality 7