a simple class of asymptotically optimal quantizers a simple class of quantizers is introduced which are asymptotically optimal as the number of quantization levels increases to infinity with respect to a mean r th power absolute error distortion measure these asymptotically optimal quantizers are very easy to compute their performance is evaluated for several distributions and compares favorably with the performance of the optimal quantizers in all cases for which the latter have been computed in addition their asymptotic robustness is studied under location scale and shape mismatch for several families of distributions

combining image compression and classification using vector quantization we describe a method of combining classification and compression into a single vector quantizer by incorporating a bayes risk term into the distortion measure used in the quantizer design algorithm once trained the quantizer can operate to minimize the bayes risk weighted distortion measure if there is a model providing the required posterior probabilities or it can operate in a suboptimal fashion by minimizing the squared error only comparisons are made with other vector quantizer based classifiers including the independent design of quantization and minimum bayes risk classification and kohonen s lvq a variety of examples demonstrate that the proposed method can provide classification ability close to or superior to learning vq while simultaneously providing superior compression performance

vector quantization technique for nonparametric classifier design an effective data reduction technique based on vector quantization is introduced for nonparametric classifier design two new nonparametric classifiers are developed and their performance is evaluated using various examples the new methods maintain a classification accuracy that is competitive with that of classical methods but at the same time yields very high data reduction rates

the graded witt ring and galois cohomology ii a primary problem in the theory of quadratic forms over a field f of characteristic different from two is to prove that the rings hq f and gwf are isomorphic here hqf h gal fq f z 2z where fq is the quadratic closure of f and gwf is the graded witt nrng associated to the fundamental ideal of even dimensional forms in the witt ring wf of f in this paper we assume we are given a field extension k of f such that wk is close to wf or hq k is close to hq f a method is developed to obtain information about these graded rings over f and its 2 extensions from information about the corresponding graded ring of k this relative theory extends and includes the previously developed absolute case where k fq applications are also given to show that hq f and gwf are isomorphic for a collection of fields anrsing naturally from the theory of abstract witt rings

symmetric bilinear forms quadratic forms and milnor k theory in characteristic two is bijective for any field f of characteristic two and for any n 0 here k f is the k group of milnor defined in 11 and i is the unique maximal ideal of the witt ring w f of non degenerate symmetric bilinear forms over f on the other hand sah studied quadratic forms in characteristic two in 15 let f be a field of characteristic two wq f the witt group of non degenerate quadratic forms over f and define i wq f n o by the canonical w f module structure on wq f then he proved that wq f iwq f is isomorphic to the group of quadratic cyclic extensions of f and iwq f i2wq f is isomorphic to the subgroup br f 2 of the brauer group br f of f consisting of elements annihilated by 2 in this paper we prove the above milnor s conjecture and describe the precise structures of w f and wq f generalizing the result of sah for any field f of characteristic p 0 let be the n th exterior power over f of the absolute differential module n t g21 let d 2 1 be the f z f fp image of the exterior derivation d f2 t f2 and let v n v resp h 1 f be the kernel resp cokernel of the homomorphism

bennett s integral for vector quantizers this paper extends bennett s 1948 integral from scalar to vector quantizers giving a simple formula that expresses the rth power distortion of a many point vector quantizer in terms of the number of points point density function inertial profile and the distribution of the source the inertial profile specifies the normalized moment of inertia of quantization cells as a function of location the extension is formulated in terms of a sequence of quantizers whose point density and inertial profile approach known functions as the number of points increase precise conditions are given for the convergence of distortion suitably normalized to bennett s integral previous extensions did not include the inertial profile and consequently provided only bounds or applied only to quantizers with congruent cells such as lattice and optimal quantizers the new version of bennett s integral provides a framework for the analysis of suboptimal structured vector quantizers it is shown how the loss in performance of such quantizers relative to optimal unstructured ones can be decomposed into point density and cell shape losses as examples these losses are computed for product quantizers and used to gain further understanding of the performance of scalar quantizers applied to stationary memoryless sources and of transform codes applied to gaussian sources with memory it is shown that the short coming of such quantizers is that they must compromise between point density and cell shapes

rigid elements valuations and realization of witt rings to the theory over the residue class field of the valuation this theorem generalizes to any 2 henselian valuation cf kl sect 12 21 let v be the valuation let a e k f o 1 be such that v a is not divisible by two the key idea behind the proof is that the quadratic form 1 a only represents elements in u a such an element a is called

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

witt rings and brauer groups under multiquadratic extensions ii let f be a field of characteristic not 2 and suppose m is a multiquadratic extension of f that is m f is a finite abelian extension of exponent 2 so that m f for some finite subgroup g e f f where f 0 in studying brauer groups and products of quaternion algebras the second author was led to consider the homology groups n m f of a certain complex gm f associated to the extension m f this complex appears in 11 3 1 301 and in 1 1 below the purpose of the present work is to investigate the first homology group n m f and to exhibit its close connections with quadratic form theory the higher homology groups are examined in 11 301 the field f is said to have property pi n if n m f 1 for every multiquadratic extension m f with m f q 2 properties p l and p 2 always hold but there are examples 29 of fields f for which p 3 fails these examples are generalized in section 5

histogram regression estimation using data dependent partitions we establish general sufficient conditions for the l 2 conoistency of multivariate histogram regression estimates based on data dependent partitions these same conditions insure the consistency of partitioning regression estimates based on local polynomial fits and with an additional regularity assumption the consistency of histogram estimates for conditional medians our conditions require shrinking cells subexponential growth of a combinatorial complexity measure and sublinear growth of restricted cell counts it is not assumed that the cells of every partition be rectangles with sides parallel to the coordinate axis or that each cell contain a minimum number of points response variables are assumed to be bounded throughout our results may be applied to a variety of partitioning schemes we established the consistency of histograms regression estimates based on cubic partitions with data dependent offsets k thresholding in one dimension and empirically optimal nearest neighbor clustering schemes in addition it is shown that empirically optimal regression trees are consistent when the size of the trees grows with the number of samples at an appropriate rate

correction to asymptotic properties of the eulerian truncation approximation analysis of the perfectly stratified transport problem by shu guang li and dennis mclaughlin

recursive partitioning to reduce distortion adaptive partitioning of a multidimensional feature space plays a fundamental role in the design of data compression schemes most partition based design methods operate in an iterative fashion seeking to reduce distortion at each stage of their operation by implementing a linear split of a selected cell the operation and eventual outcome of such methods is easily described in terms of binary tree structured vector quantizers this paper considers a class of simple growing procedures for tree structured vector quantizers of primary interest is the asymptotic distortion of quantizers produced by the unsupervised implementation of the procedures it is shown that application of the procedures to a convergent sequence of distributions with a suitable limit yields quantizers whose distortion tends to zero analogous results are established for tree structured vector quantizers produced from stationary ergodic training data the analysis is applicable to procedures employing both axis parallel and oblique splitting and a variety of distortion measures the results of the paper apply directly to unsupervised procedures that may be efficiently implemented on a digital computer

versions of springer s theorem for quadratic forms in characteristic 2 whenever f is a henselian valued field whose residue class field inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i has characteristic different from 2 a theorem of springer completely reduces the quadratic form theory of f to the quadratic form theory of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i and the structure of the value group this paper presents analogues of this result in the case where f is a maximally complete field of characteristic 2 two different results are given the first determines the witt group and the second classifies the anisotropic quadratic forms

vanishing distortion and shrinking cells we establish an asymptotic connection between vanishing rth power distortion and shrinking cell diameters for vector quantizers with convex cells as a consequence a number of shrinking cell conditions may be easily verified by showing that the quantizers in question have distortion tending to zero this also plays an important role in the asymptotic analysis of a common greedy growing scheme for tree structured vector quantizers

on the asymptotic distribution of the errors in vector quantization in a recent paper lee and neuhoff see ibid vol 42 no 2 p446 60 1996 found an asymptotic formula for the distribution of the length of the errors produced by a vector quantizer with many quantization points this distribution depends on the source probability density the quantizer point density and the quantizer shape profile the latter characterizes the shapes of the quantization cells as a function of position the purpose of this paper is to give a rigorous derivation of this formula by identifying precise conditions under which it is shown that if a sequence of vector quantizers with a given dimension and an increasing number of points has specific point densities and specific shape profiles converging to a model point density and a model shape profile respectively then the distribution of the length of the quantization error suitably normalized converges to the aforementioned formula with the model point density and the model shape profile substituted