Block diagrams of two-stages selleck compound residual vector quantization. (a) Learning codebooks; (b) Quantizing a vector.For L stages residual vector quantization, a vector x is approximated by the sum of its L stages�� quantization outputs while the last stage��s quantization error is discarded:x=��i=1Lx?i+?L�֡�i=1Lx?i=x?(5)For transformation or storage, indices of quantization outputs are used. For L stage residual vector quantization, which is constructed by K-point vector quantizers, the bit rate is L log2 K per vector.The quantization performance of ith stage-quantizer is:MSE(Qi)=1N��?��Ei?T?=1N��j=1K��x��Vj��x?ci,j��2(6)where Inhibitors,Modulators,Libraries Inhibitors,Modulators,Libraries Ei is the new residual vector set generated by Qi, Vj is the jth cluster and ci,j is Vj��s centroid.

Considering the optimization problem of finding a vector y to minimize the objection function:J=��x��Vj��x?y��2(7)By differentiating the objection function J with respect Inhibitors,Modulators,Libraries to y and setting derivative equal to zero, it is easy to obtain the minimizing y:y=1Nj��x��Vjx(8)where Nj is the number of vectors in jth cluster. This means the centroid of cluster minimizes the objection function:��x��Vj��x?ci,j��2=miny��x��Vj��x?y��2�ܡ�x��Vj��x?y��2|y=0=��x��Vj��x��2(9)With the observations that ��?��Ei?T?=��j=1K��x��Vj��x?ci,j��2 and ��x��Ei?1xTx=��j=1K��x��Vj��x��2, we obtain the inequality:MSE(Qi)��MSE(Qi?1)(10)which Inhibitors,Modulators,Libraries means the k-means clustering method guarantee the MSE of stage-quantizers are decreasing monotonically.3.?Using Residual Vector Quantization for ANN3.1. Exhaustive Search by Fast Distance ComputationIn [17] the exact Euclidean distance between two vectors is approximated by asymmetric distance, i.

e., the distance between a vector and a reproduction Brefeldin_A of another vector:d(x,?y)��d?(x,?y)=d(x,?Q(y))(11)Asymmetric distance reduces the quantization noise and improves the search quality [17]. We have proposed fast asymmetric distance computation based on residual vector quantization. Suppose a database vector y is quantized by L �� K residual vector quantizer, its indices of quantization output are uj, 1 �� uj �� K, j =1..L, and the reproduction of y is constructed by the sum of MEK162 msds corresponding centroids:y?=��i=1Ly?i=��i=1Lci,ui,?ci,ui��Ci,?1��ui��K(12)where ci,ui is the uith centroid of codebook Ci. The squared asymmetric distance between y and the target vector x is the exact squared distance between x and ?:d?(x,?y)2=d(x,?y?)2=��x?y?��2=��x��2+��y?��2?2?x,?y??=��x��2+��y?��2?2?x,?��i=1Lci,ui?=��x��2+��y?��2?2��i=1L?x,?ci,ui?(13)where x, y is dot product. ||?|| is pre-computed off-line when the database vector is quantized.