Abstract:
An adaptive pattern classifier makes use of training patterns and a known non-linear invariance transformation to generate a classifier representation based on an infinite set of virtual training samples on a training trajectory. Given the non-linear invariance transformation, optimization can be formulated as a semidefinite program (SDP), which is given by a linear objective function that is minimized subject to a linear matrix inequality (LMI). In this manner, a small training set may be virtually supplemented using the non-linear invariance transformation to learn an effective classifier that satisfactorily recognizes patterns, even in the presence of known transformations that do not change the class of the pattern.

Description:
TECHNICAL FIELD  
       [0001]     The invention relates generally to learning algorithms, and more particularly to invariant pattern recognition using semidefinite programming techniques.  
       BACKGROUND  
       [0002]     Pattern recognition has been applied to a variety of problems, including handwriting recognition, speech recognition, image understanding, etc. A central problem of pattern recognition is the exploitation of known invariants in a pattern domain. Invariance represents a difference in individual target patterns (e.g., images) of a given classification that does not alter their recognized classification. For example, given two different instances of a handwritten numeral ‘9’ (see patterns  100  and  102  in  FIG. 1 ), both instances are still recognized (or classified) as a numeral ‘9’. One approach to handling invariants is to introduce invariant information into the pattern classifier&#39;s training set, which is used to generate the classifier.  
         [0003]     In images, known invariants may include rotation, translation, sheer, scaling, brightness and lighting direction. In addition, in handwriting recognition, invariants may include line thinning/thickening and other non-uniform character differences. Also, in speech recognition (e.g., phoneme classification), invariants may include differences in base frequency or duration. A wide variety of other invariants, singly or in combination, are also possible in a wide assortment of applications. When using machine learning approaches in these applications, it is a challenge to combine a training set with knowledge of invariants in order to obtain a satisfactory pattern classifier, particularly when the training set is small.  
         [0004]     One existing method of incorporating invariants is to include discrete invariant samples in the training set. For example, one approach takes multiple samples of a handwritten numeral ‘9’ in the presence of invariants and adding them to the training set. Another approach generates discrete “virtual” invariant samples by the application of an invariance transformation and adds these virtual samples to the training set. However, these approaches can result in inconvenient or unwieldy training sets and, nevertheless, can result in incorrect classification caused by gaps between the discrete training samples.  
         [0005]     Another method involves modeling invariants using a linear approximation. However, linear approximations of invariants are substantially inaccurate in most circumstances. Nevertheless, while non-linear approximations may provide greater accuracy, existing approaches have been unable to solve for non-linear approximations within given constraints.  
       SUMMARY  
       [0006]     Implementations described and claimed herein address the foregoing problems using a non-linear invariant transformation to generate an infinite set of virtual training samples on a training trajectory. Given the non-linear invariant transformation, optimization can be formulated as a semidefinite program (SDP), which is given by a linear objective function that is minimized subject to a linear matrix inequality (LMI). In this manner, a small training set may be virtually supplemented using the non-linear invariant transformation to learn an effective linear classifier that satisfactorily recognizes patterns, even in the presence of invariant differences in the patterns.  
         [0007]     In some implementations, articles of manufacture are provided as computer program products. One implementation of a computer program product provides a computer program storage medium readable by a computer system and encoding a computer program. Another implementation of a computer program product may be provided in a computer data signal embodied in a carrier wave by a computing system and encoding the computer program.  
         [0008]     The computer program product encodes a computer program for executing on a computer system a computer process. Derivatives of a nonlinear invariance transformation at a training data point are generated with respect to a transformation parameter. The training data point represents one of a plurality of training patterns. A classifier representation is generated for classifying a test pattern in the presence of the nonlinear invariance transformation.  
         [0009]     In another implementation, a method is provided. Derivatives of a nonlinear invariance transformation at a training data point are generated with respect to a transformation parameter. The training data point represents one of a plurality of training patterns. A classifier representation is generated for classifying a test pattern in the presence of the nonlinear invariance transformation.  
         [0010]     In another implementation, a system includes a derivative generator that generates derivatives of a nonlinear invariance transformation at a training data point with respect to a transformation parameter. The training data point represents one of a plurality of training patterns. A classifier representation generator generates a classifier representation for classifying a test pattern in the presence of the nonlinear invariance transformation.  
         [0011]     Other implementations are also described and recited herein. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0012]      FIG. 1  illustrates a numeral pattern and an exemplary invariant pattern.  
         [0013]      FIG. 2  illustrates a graph including exemplary six training objects and associated invariant transformation trajectories.  
         [0014]      FIG. 3  illustrates a graph including six exemplary training objects and associated virtual invariant transformation trajectories.  
         [0015]      FIG. 4  shows a graph of data points and virtual invariant transforms for exemplary objects.  
         [0016]      FIG. 5  illustrates an exemplary pattern recognition system useful for invariant pattern recognition.  
         [0017]      FIG. 6  illustrates exemplary operations for performing invariant pattern recognition.  
         [0018]      FIG. 7  illustrates a system useful for implementing an embodiment of the present invention. 
     
    
     DETAILED DESCRIPTION  
       [0019]      FIG. 1  illustrates a numeral pattern and an exemplary invariant pattern. Numeral pattern  100  represents a training pattern input as a member of an initial training set. The numeral pattern  100  is classified as a ‘9’. The exemplary invariant pattern  102 , while differing from numeral pattern  100  by virtue of at least a slight rotation, is also classified as a ‘9’. Therefore, despite the differences between the patterns  100  and  102 , the classifications are the same (i.e., invariant). In the illustrated implementation, the patterns  100  and  102  are shown as a set of pixels, where each pixel is described by an intensity value (e.g., selected 255 possible gray values). Given such intensity values, the numeral pattern  100  is characterized for inclusion in the training set.  
         [0020]     Various methods of characterizing the patterns may be employed. In one implementation, the mean pixel intensity of all pixels within each of the three regions ( 104 ,  106 , and  108 ) is computed, as shown by feature values φ 1 , φ 2 , and φ 3  (i.e., values  110 ,  112 , and  114 , respectively). It should be understood that alternative methods of characterizing a pattern may be employed, including without limitation those that incorporate colors, individual pixel values, different regions, statistical analysis, etc. In addition, for non-visual patterns, characterization may depend on other parameters, such as frequency composition, amplitude, timing, or some other audio parameters for speech patterns. The invariant pattern  102  is also shown as characterized by feature values  116 , demonstrating how the feature values of patterns in the same class can differ.  
         [0021]      FIG. 2  illustrates a graph  200  including six exemplary training objects, wherein each training object is characterized by two feature values. For example, the training object may be a graphical training pattern. The x-axis  202  of the graph  200  represents possible values of the first feature value φ 1  of each training object, and the y-axis  204  represents possible values of the second feature value φ 2  of each training object. A two-dimensional graph  200  is used herein only in order to facilitate description. It should be understood, however, that objects may be characterized by any number of feature values.  
         [0022]     Each illustrated training object  206 ,  208 , and  210  (representing ‘4’) and  212 ,  214 , and  216  (representing ‘8’) is represented by a vector x i , where each vector component represents a feature value φ j , j represents the index of the feature value, and i represents the index distinguishing each training object (e.g., x i =(φ 1  φ 2 )). For example, the training object  206  may be represented by x 1 =(0.144 0.349), and the training object  208  may be represented by x 2 =(0.188 0.399).  
         [0023]     In the proximity of each object data point (e.g., data point  218  associated with object  212 ), data points of possible invariant patterns (e.g., invariant data points  220 ) associated with the corresponding object are shown. In the illustration, the invariants are the result of rotation within a predefined degree range (e.g., ±12°), although other invariants may also be present. It should be noted that the curves corresponding to transformed data points associated with several of the training objects are substantially non-linear. As such, modeling the invariant transforms of each object using a polynomial of degree r≧2 can improve classification accuracy over linear approaches. An invariant transformation can model a variety of invariants, including pattern rotations, translations, sheering, scaling, brightness, lighting direction, line thinning/thickening, frequency differences, timing differences, etc.  
         [0024]     A conceptual objective of a linear classifier is to determine a hyper-plane (e.g., a line  222  in two dimensions) that distinctly separates objects of different classes and maximizes equal margins (e.g., margins  224 ) on either side of the line. Objects on one side of the line are in one class; objects on the other side of the line are in another class. The margin is maximized to provide the greatest probability of correctly classifying invariant test objects that may be input to the classifier. Since the features φ j  used can be arbitrary functions of the original input representation of the training objects including polynomials and radial basis functions the linear classifier above can implement classification that is non-linear in the original input representation.  
         [0025]      FIG. 3  illustrates a graph  300  including six exemplary training objects and associated virtual invariant transformation trajectories. As can be seen clearly with regard to objects  302  and  304 , non-linear modeling (e.g., shown as dotted line  306  for object  304 ) more closely approximates the actual invariant transformations of each object (e.g., shown as solid line  308  for object  304 ) than linear modeling.  
         [0026]      FIG. 4  shows a graph  400  of data points and virtual invariant transforms for exemplary training objects. A magnified portion of the graph  400  is shown in graph  402 . A training object data point  404  is associated with discrete virtual invariant data points  406 ,  408 ,  410 ,  412 , and  414  which may be generated based on a polynomial transformation. It can be seen, however, that the original training object data point  404  and the discrete virtual data points  406 ,  408 ,  410 ,  412 , and  414  may be insufficient to provide an optimal solution if the region on the invariant transformation trajectory  416  between data points  404  and  414  falls into the margin region, resulting in possible misclassifications. As such, considering the invariant transformation trajectory  416  itself and aligning the margin  418  tangent to the trajectory  416 , as described herein, provides a more accurate linear classifier.  
         [0027]     As previously described, a training object set ((x 1 ,y i ), . . . , (x m ,y m ))ε(           n ×{−1,+1}) m  may be obtained from training pattern input data, where x i  represents a vector characterizing a given training pattern and y i  represents a binary classification for the given training pattern (e.g., as representing either a ‘4’ or an ‘8’ in  FIG. 3 ). For example, in  FIG. 3 , three patterns of handwritten ‘4’s and three patterns or handwritten ‘8’s are received as training data input and, based on the training object&#39;s location relative to the line  310  each training object is classified as either a four or an eight.  
         [0028]     The classification of test patterns x by y(x)=sign (w T x) is based on a weight vector Wε           n . Assuming linear separability of the training patterns, the principle of empirical risk minimization recommends finding a weight vector w such that, for all iε{1 . . . , m}, y i w T x i ≧0. As such, the problem constitutes a linear feasibility problem.  
         [0029]     However, in order to correctly classify patterns under the known invariance transformation T(θ,·), the approach described above is generalized into the following feasibility problem: 
 
find wε           n  such that ∀iε{1, . . . , m}: ∀θε         : y i w T x i (θ)≧0   (1) 
 
 That is, the weight vector would correctly classify every transformed test pattern x i (θ):=T(θ, x i ) for every value of the transformation parameter θ. 
 
         [0030]     The invariance transformation T can be approximated by a transformation polynomial in θ, such as a Taylor expansion of the form:  
                   T   ^     ⁡     (     θ   ,     x   i       )       ≈       T   ⁡     (     0   ,     x   i       )       +       ∑     j   =   1     r     ⁢           ⁢       θ   j     ·     (         1     j   !       ⁢         ⅆ   j     ⁢     T   ⁡     (     θ   ,     x   i       )           ⅆ     θ   j           ⁢     ❘     θ   =   0         )             ⁢     
     ⁢           =       T   ⁡     (     0   ,     x   i       )       +       ∑     j   =   1     r     ⁢           ⁢       θ   j     ·       (     X   i     )       j   ,   .                     (   2   )             
 
 However, other forms of transformation polynomial may be employed. 
 
         [0031]     Given this approximation, the feasibility problem may be simplified by considering only the transformations on the trajectory 
 
{{tilde over (T)}(θ,x): θε         }⊂           n    (3) 
 
 of polynomial form (i.e., {tilde over (x)} i (θ):={tilde over (T)}(θ,x i )=X i   T θ), each polynomial example {tilde over (x)} i (θ) being represented by a polynomial and its derivatives in the r+1 row vectors of X i ε           r+1)×n , with θ:=(1, θ, . . . , θ r ) T . The columns j of X i  represent the coefficients of the polynomial and its derivatives (e.g., x′,x″). For example where r=2:  
               X   i     =     (         x             x   ′               x   ″           )             (   4   )             
 
 The feasibility problem defined by Equation (1) thus simplifies to: 
 
find wε           n  such that ∀iε{1, . . . , m}: ∀θε         : y i w T X i   T θ≧0   (5) 
 
 which represents finding a weight vector w such that the polynomials p i (θ)=y i w T X i   T θ are non-negative everywhere (i.e., p i εP r   + ). 
 
         [0032]     Given the problem statement defined in Equation (5), the following proposition by Nesterov supports a semidefinite programming formulation of Equation (5), whenever r=2l:  
         [0033]     Seinidefinite Representation of Non-Negative Polynomials (Nesterov)  
         [0034]     The set P 2l   +  of polynomials that are non-negative everywhere on the real line is semidefinite representable in the sense that 
        1. for every positive semidefinite matrix P≧0, the polynomial p(θ)=θ T Pθ is non-negative everywhere, pεP 2l   + .     2. for every polynomial pεP 2l   + , there exists a positive semidefinite matrix P≧0 such that p(θ)=θ T Pθ.        
 
         [0037]     In other words, Nesterov proposes that the set of polynomials P 2l   +  of degree 2l non-negative on the entire real line is a convex set representable by positive semidefinite (psd) constraints. Therefore, optimization over P 2l   +  can be formulated as a semidefinite program (SDP), which may be given by a linear objective function that is minimized subject to a linear matrix inequality (LMI):  
                 minimize     w   ∈     ℝ   n         ⁢           ⁢     c   T     ⁢   w   ⁢           ⁢   subject   ⁢           ⁢   to   ⁢           ⁢     A   ⁡     (   w   )         :=           ∑     k   =   1     n     ⁢           ⁢       w   k     ⁢     A   k         -   B     ≽   0             (   6   )             
 
 The LMI A(w)≧0 means that A(w) is required to be positive semidefinite. That is, for all vε           n ,  
                 v   T     ⁢     A   ⁡     (   w   )       ⁢   v     =           ∑     k   =   1     n     ⁢           ⁢       w   k     ⁡     (       v   T     ⁢     A   k     ⁢   v     )         -       v   T     ⁢   Bv       ≥   0             (   7   )             
 
 which reveals that LMI constraints correspond to infinitely many linear constraints. This expressive power can be used to enforce constraints for training objects as given by Equation (3) (i.e., constraints can be used to hold for all values θε         ) Based on the representability theorem for non-negative polynomials, a learning algorithm can be developed in the form of a Semidefinite Programming Machine (SDPM) that maximizes the margin on polynomial training objects, much like the support vector machine for ordinary single vector data. 
 
         [0038]     As such, Nesterov&#39;s proposition may be applied to develop a semidefinite programming formulation for the problem of learning a maximum margin classifier given the polynomial constraints of Equation (5). In one implementation, the squared objective ∥w∥ 2  is minimized by replacing it with an auxiliary variable t, subject to a quadratic constraint t≧∥w∥ 2  that is written as an LMI (i.e., an optimization problem) using Schur&#39;s complement lemma:  
                     minimize     (     w   ,   t     )       ⁢     1   2     ⁢   t   ⁢           ⁢   subject   ⁢           ⁢   to   ⁢           ⁢     F   ⁡     (     w   ,   t     )         :=       (           I   n         w             w   T         t         )     ≽   0       ,   and     ⁢     
     ⁢       ∀     i   :     G   ⁡     (     w   ,     X   i     ,     y   i       )           :=         G   0     +       ∑     k   =   1     n     ⁢           ⁢       w   k     ⁢       G   k     ⁡     (         (     X   i     )       .     ,   k         ,     y   i       )             ≽   0               (   8   )             
 
 where (X i ) ·,k  denotes the k th  column of X i  and G≧0 represents a semidefinite programming constraint. Therefore, Equation (8) represents a semidefinite programming problem. 
 
         [0039]     For illustration, consider the case of l=0 (the simplest non-trivial case). The constraint matrix G(w,X i ,y i ) reduces to a scalar y i w T x i −1, which translates into the standard support vector machine constraint y i w T x i ≧1 linear in w.  
         [0040]     For l=1, G(w, X i ,y i )ε           2×2  and  
               G   ⁡     (     w   ,     X   i     ,     y   i       )       :=     (               y   i     ⁢         w   T     ⁡     (     X   i     )         0   ,   .         -   1             1   2     ⁢     y   i     ⁢         w   T     ⁡     (     X   i     )         1   ,   .                     1   2     ⁢     y   i     ⁢         w   T     ⁡     (     X   i     )         1   ,   .                 y   i     ⁢         w   T     ⁡     (     X   i     )         2   ,   .               )             (   9   )             
 
 where (X i ) 0,·  represents the original data vector, (X i ) 1,·  represents the first derivative of the transformation polynomial at θ=0, and (X i ) 2,·  represents the second derivative of the transformation at θ=0. 
 
         [0041]     For the case of l≧2, the resulting problem constitutes a full semidefinite program. For illustration, consider the case l=2. Because a polynomial p of degree four is fully determined by its five coefficients p 0 , . . . , p 4 , but the symmetric matrix Pε           3×3  in p(θ)=θ T Pθ has six degrees of freedom, one auxiliary variable u i  is used per training object:  
               G   ⁡     (     w   ,     u   i     ,     X   i     ,     y   i       )       :=     (               y   i     ⁢         w   T     ⁡     (     X   i     )         0   ,   .         -   1             1   2     ⁢     y   i     ⁢         w   T     ⁡     (     X   i     )         1   ,   .                     y   i     ⁢         w   T     ⁡     (     X   i     )         2   ,   .         -     u   i       2                 1   2     ⁢     y   i     ⁢         w   T     ⁡     (     X   i     )         1   ,   .               u   i             1   2     ⁢     y   i     ⁢         w   T     ⁡     (     X   i     )         3   ,   .                         y   i     ⁢         w   T     ⁡     (     X   i     )         2   ,   .         -     u   i       2             1   2     ⁢     y   i     ⁢         w   T     ⁡     (     X   i     )         3   ,   .                 1   2     ⁢     y   i     ⁢         w   T     ⁡     (     X   i     )         4   ,   .               )             (   10   )             
 
 Generally, because a polynomial of degree 2l has 2l+1 coefficients and a symmetric (l+1)×(l+1) matrix has  
         1   2     ⁢     (       (     l   +   1     )     ×     (     l   +   2     )       )         
 
 degrees of freedom, then  
         1   2     ⁢     (     l   -   1     )     ⁢   l       
 
 auxiliary variables are used. 
 
         [0042]     The dual of the general semidefinite program (shown in Equation (5) is represented by:  
                     maximize     Λ   ∈     ℝ   mxm         ⁢           ⁢     tr   ⁡     (     B   ⁢           ⁢   Λ     )       ⁢           ⁢   subject   ⁢           ⁢   to   ⁢           ⁢     ∀     j   ∈       {     1   ,   …   ⁢           ,   n     }     ⁢     :     ⁢     tr   ⁡     (       A   j     ⁢   Λ     )               =     c   j       ;     ⁢     
     ⁢     Λ   ≽   0             (   11   )             
 
 where Λ represents a matrix of dual variables. 
 
         [0043]     The complementarity conditions for the optimal solution (w*,t*) read A((w*,t*))Λ*=0. Therefore, the dual formulation of Equations (8) and (9) for l=1, combined with the F(w,t) portions of the complementarity conditions may be given by:  
                     maximize       (     α   ,   β   ,   γ     )     ∈     ℝ     3   ⁢   m           -       1   2     ⁢       ∑     j   =   1     m     ⁢           ⁢       ∑     k   =   1     m     ⁢       y   j     ⁢           y   k     ⁡     [       x   ~     ⁡     (       α   j     ,     β   j     ,     γ   j     ,     X   j       )       ]       T     ⁡     [       x   ~     ⁡     (       α   k     ,     β   k     ,     γ   k     ,     X   k       )       ]               +                       ⁢       ∑     j   =   1     m     ⁢           ⁢     α   j                       subject   ⁢           ⁢   to   ⁢           ⁢     ∀     j   ∈     {     1   ,   …   ⁢           ,   m     }           :     M   j       :=       (           α   j           β   j               β   j           γ   j           )     ≽   0                   (   12   )             
 
 with extrapolated training objects {tilde over (x)}(α i ,β i ,γ i ,X i ):=α i (X i ) 0,· +β i (X i ) 1,· +γ i (X i ) 2,· . Equation (9) has a quadratic objective and positive semidefinite constraints and can therefore be formulated as a standard semidefinite program in the form of Equation (6). In addition, the complementarity conditions reveal that the optimal weight vector w* can be expanded as:  
               w   *     =       ∑     i   =   1     m     ⁢           ⁢       y   i     ⁢       x   ~     ⁡     (       α   i     ,     β   i     ,     γ   i     ,     X   i       )                   (   13   )             
 
         [0044]     Given l=1, using Equation (6) and assuming primal and dual feasibility, 
 
 G ( w*,X   i   ,y   i )· M*   i =0   (14) 
 
 for all iε{1, . . . , m} at the solution (w*,t*,M* i ). The trace of Equation (14) translates into: 
 
 y   i   w   T (α* i ( X   i ) 0,· +β* i ( X   i ) 1,· +γ* i ( X   i ) 2,· )=α* i    (15) 
 
         [0045]     The following proposition allows characterization of the solution:  
         [0000]     Sparse Expansion  
         [0046]     The expansion in Equation (13) of the optimal weight vector w* in terms of training objects X i  is sparse in the following sense:  
         [0047]     Only those training objects X i  (“support vectors”) may have non-zero expansion coefficients α* i , which lie on the margin (i.e., for which G i (w*,X i ,y i )≧0 rather than G i (w*,X i ,y i )&gt;0. Furthermore, in this case, α* i =0 implies β* i =γ* i 0 as well.  
         [0048]     In addition, the expansion of Equation (13) is further characterized by the following proposition:  
         [0000]     Truly Virtual Support Vectors  
         [0049]     For all training objects represented by X i  lying on the margin (i.e., satisfying G i (w*,X i ,y i )≧0 and M* i ≧0), there exists θ i ε         ∪{∞} such that the optimal weight vector w* can be written as  
               w   *     =         ∑     i   =   1     m     ⁢           ⁢       α   i   *     ⁢     y   i     ⁢         x   ~     i     ⁡     (     θ   i     )           =       ∑     i   =   1     m     ⁢           ⁢       y   i     ⁢       α   i   *     ⁡     (         (     X   i     )       0   ,   *       +         θ   i   *     ⁡     (     X   i     )         1   ,   *       +         θ   i     *   2       ⁡     (     X   i     )         2   ,   *         )                     (   16   )             
 
         [0050]     Based on Equation (16), it is thus possible to identify both (1) the training objects represented by X i  that are used in the expansion of the optimal weight vector w* and (2) the corresponding values θ* i  of the transformation parameter θ. Therefore, an exemplary semidefinite programming machine may be used to find virtual support vectors that were not explicitly provided in the original training object set. The result induces a classifier: 
 
 y ( x )=sign(( w* ) T   x )   (17) 
 
         [0051]      FIG. 5  illustrates an exemplary pattern recognition system  500  useful for invariant pattern recognition. Training data  502  is input to a training data characterizer  504 , which characterizes individual training patterns on the basis of feature values in a given number of dimensions r (e.g., in  FIG. 1 , three dimensions are shown; in  FIG. 2 , two dimensions are shown). Data points in n-dimensional feature space are defined by the feature values of each training pattern. Each training pattern is also associated with its classification (e.g., y i ).  
         [0052]     A derivative generator  506  computes derivatives of a non-linear invariance transformation at each data point characterized by the training data characterizer  504 . As described above, a polynomial of the form identified in Equation (2) may be used to approximate the invariant transformation. The coefficients of the transformation and its derivatives at each data point are input to a classifier representation generator  508 , which outputs a classifier representation  510 . In one implementation, the classifier representation  510  is in the form of an optimized weight vector w*, as shown in Equation (16).  
         [0053]     In some implementations, a test pattern  512  (e.g., a new handwritten character to be classified) may be input to a classifier  514 , which, based on the input classifier representation  510 , classifies the test pattern  512 , such as by applying an equation of the form shown in Equation (17), to yield a classification signal  514 . In one implementation, the classification signal  514  indicates to which class (e.g., of two classes) the pattern belongs.  
         [0054]     It should be understood, however, that some implementations of the system  500  may be developed without the classifier  514 . In these implementations, an optimized classifier representation may be generated and stored or distributed for future use with the classifier  514  or some other classifier.  
         [0055]     As shown, in some implementations, the test pattern  512  and the classification signal  514  may be fed back to the derivative generator  506  to tune the classifier representation  510  based on the newly classified test pattern  512 .  
         [0056]      FIG. 6  illustrates exemplary operations  600  for performing invariant pattern recognition. A receiving operation  602  receives training pattern data. A characterization operation  604  characterizes the training patterns in a given number of dimensions n by producing n feature values for each pattern. Each training pattern is also associated with its classification (e.g., y i ). A derivative operation  606  generates derivatives of a non-linear invariant transformation at each data point defined by the feature values of each pattern. Using the set of coefficients of the non-linear transformation and its derivatives at each data point, a classifier representation is generated at generation operation  608 . In one implementation, the classifier representation is in the form of an optimized weight vector w*, as shown in Equation (16).  
         [0057]     Another receiving operation  610  receives a test pattern, and a classifying operation  612  classifies the test pattern based on the classifier representation. For example, the classifying operation  612  may apply an equation of the form shown in Equation (17), to yield a classification signal in a generation operation  614 . In one implementation, the classification signal indicates to which class (e.g., of two classes) the pattern belongs.  
         [0058]     It should be understood that a classifier representation may be generated without proceeding to the classification operation. In these implementations, an optimized classifier representation may be generated and stored or distributed for future use with a classifier.  
         [0059]     In one implementation, a feedback operation  616  may input the test pattern and the associated classification signal to the derivative operation  606  in order to refine the classifier representation.  
         [0060]     In other implementations, it may not be desirable to enforce correct classification on the entire trajectory given by the polynomial example {tilde over (x)}(θ). In particular, when the polynomial is used as a local approximation to a global invariance, the polynomial may be restricted to a segment of the trajectory. Therefore, based on the following corollary to the Nesterov Proposition given herein, the examples {tilde over (x)}(θ) may be restricted to a closed interval θε[−τ,τ] on the real line by effectively doubling the degree of the polynomial used:  
         [0061]     SD-Representability on a Segment  
         [0062]     The set P i   +  (−τ,τ) of polynomials non-negative on a segment [−τ,τ] is SD-representable.  
         [0000]     Note that the matrix G(w,X i ,y i ) is sparse because the resulting polynomial contains even powers of θ.  
         [0063]     In another implementation, more than one transformation may be considered concurrently. For example, in handwritten digit recognition, transformations such as rotation, scaling, translation, shear, thinning/thickening, etc. may all be relevant. The first statement of the Nesterov Proposition may be generalized to polynomials of more than one variable: for every psd matrix P≧0, the polynomial p(θ)=v θ   T Pv θ  is non-negative everywhere, even if v i  is any power of θ j . Therefore, optimization is only over a subset of these polynomials.  
         [0064]     For example, polynomials of degree two and θ:={1,θ 1 , . . . , θ D }.  
                 x   ~     ⁡     (   θ   )       ≈         θ   T     ⁡     [             x   i     ⁡     (   0   )               ∇   θ   T     ⁢       x   i     ⁡     (   0   )                     ∇   θ     ⁢       x   i     ⁡     (   0   )                 ∇   θ     ⁢       ∇   θ   T     ⁢       x   i     ⁡     (   0   )                 ]       ⁢   θ             (   18   )             
        where ∇ θ   T  represents the gradient and ∇ θ ∇ θ   T  represents the Hessian operator. Note that the scaling behavior is with regard to the number D of parameters.        
 
         [0066]     Support vector machines derive much of their popularity from the flexibility added by the use of kernels. Taking the dual SDPM as a starting point and assuming the Taylor expansion in Equation (2), differentiating through the kernel function allows representation of the polynomial trajectory in the feature space.  
         [0067]     Therefore, it may be assumed that a feature map φ:           n →F⊂           N  is fully characterized by k:χ×χ→         , the kernel function corresponding to φ in the sense that ∀x,{tilde over (x)}εχ:[φ(x)] T [φ({tilde over (x)})]=k(x,{tilde over (x)}). The polynomial expansion (e.g., the Taylor expansion) is carried out in F. An inner product expression between data points x i  and x j  differentiated, respectfully, u and v times reads  
               [       ϕ     (   u   )       ⁡     (     x   i     )       ]     T     ⁡     [       ϕ     (   v   )       ⁡     (     x   j     )       ]         ︸       k     (     u   ,   v     )       ⁡     (       x   i     ,     x   j       )           =       ∑     s   =   1     N     ⁢           ⁢       (           d   u     ⁢       ϕ   s     ⁡     (     x   ⁡     (   θ   )       )           d   ⁢           ⁢     θ   u         ⁢     |         x   ~     =     x   i       ,     θ   =   0           )     ·     (           d   v     ⁢       ϕ   s     ⁡     (       x   ~     ⁡     (     θ   ~     )       )           d   ⁢           ⁢       θ   ~     u         ⁢     |         x   ~     =     x   j       ,       θ   ~     =   0           )             
 
         [0068]     The exemplary hardware and operating environment of  FIG. 7  for implementing the invention includes a general purpose computing device in the form of a computer  20 , including a processing unit  21 , a system memory  22 , and a system bus  23  that operatively couples various system components, including the system memory and the processing unit  21 . There may be only one or there may be more than one processing unit  21 , such that the processor of computer  20  comprises a single central-processing unit (CPU), or a plurality of processing units, commonly referred to as a parallel processing environment. The computer  20  may be a conventional computer, a distributed computer, or any other type of computer; the invention is not so limited.  
         [0069]     The system bus  23  may be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, a switched fabric, point-to-point connections, and a local bus using any of a variety of bus architectures. The system memory may also be referred to as simply the memory, and includes read only memory (ROM)  24  and random access memory (RAM)  25 . A basic input/output system (BIOS)  26 , containing the basic routines that help to transfer information between elements within the computer  20 , such as during start-up, is stored in ROM  24 . The computer  20  further includes a hard disk drive  27  for reading from and writing to a hard disk, not shown, a magnetic disk drive  28  for reading from or writing to a removable magnetic disk  29 , and an optical disk drive  30  for reading from or writing to a removable optical disk  31  such as a CD ROM or other optical media.  
         [0070]     The hard disk drive  27 , magnetic disk drive  28 , and optical disk drive  30  are connected to the system bus  23  by a hard disk drive interface  32 , a magnetic disk drive interface  33 , and an optical disk drive interface  34 , respectively. The drives and their associated computer-readable media provide nonvolatile storage of computer-readable instructions, data structures, program modules and other data for the computer  20 . It should be appreciated by those skilled in the art that any type of computer-readable media which can store data that is accessible by a computer, such as magnetic cassettes, flash memory cards, digital video disks, random access memories (RAMs), read only memories (ROMs), and the like, may be used in the exemplary operating environment.  
         [0071]     A number of program modules may be stored on the hard disk, magnetic disk  29 , optical disk  31 , ROM  24 , or RAM  25 , including an operating system  35 , one or more application programs  36 , other program modules  37 , and program data  38 . A user may enter commands and information into the computer  20  through input devices such as a keyboard  40  and pointing device  42 . Other input devices (not shown) may include a microphone, joystick, game pad, satellite dish, scanner, or the like. These and other input devices are often connected to the processing unit  21  through a serial port interface  46  that is coupled to the system bus, but may be connected by other interfaces, such as a parallel port, game port, or a universal serial bus (USB). A monitor  47  or other type of display device is also connected to the system bus  23  via an interface, such as a video adapter  48 . In addition to the monitor, computers typically include other peripheral output devices (not shown), such as speakers and printers.  
         [0072]     The computer  20  may operate in a networked environment using logical connections to one or more remote computers, such as remote computer  49 . These logical connections are achieved by a communication device coupled to or a part of the computer  20 ; the invention is not limited to a particular type of communications device. The remote computer  49  may be another computer, a server, a router, a network PC, a client, a peer device or other common network node, and typically includes many or all of the elements described above relative to the computer  20 , although only a memory storage device  50  has been illustrated in  FIG. 7 . The logical connections depicted in  FIG. 7  include a local-area network (LAN)  51  and a wide-area network (WAN)  52 . Such networking environments are commonplace in office networks, enterprise-wide computer networks, intranets and the Internet, which are all types of networks.  
         [0073]     When used in a LAN-networking environment, the computer  20  is connected to the local network  51  through a network interface or adapter  53 , which is one type of communications device. When used in a WAN-networking environment, the computer  20  typically includes a modem  54 , a network adapter, a type of communications device, or any other type of communications device for establishing communications over the wide area network  52 . The modem  54 , which may be internal or external, is connected to the system bus  23  via the serial port interface  46 . In a networked environment, program modules depicted relative to the computer  20 , or portions thereof, may be stored in the remote memory storage device. It is appreciated that the network connections shown are exemplary and other means of and communications devices for establishing a communications link between the computers may be used.  
         [0074]     In an exemplary implementation, a training data characterizer, a derivative generator, a classifier representation generator, a classifier, and other modules may be incorporated as part of the operating system  35 , application programs  36 , or other program modules  37 . Training data, a classifier representation, a test pattern, a classification signal, and other data may be stored as program data  38 .  
         [0075]     Some exemplary test patterns have been identified as image patterns and audio patterns. Other test patterns, including time series, may also be classified. I Generally, a time series includes data ordered by the time the data were collected (usually spaced at equal intervals). Common examples of a time series include daily temperature measurements, monthly sales, and yearly population figures. Typical goals of time series analysis are to describe the process generating the data and to forecast future values.  
         [0076]     The embodiments of the invention described herein are implemented as logical steps in one or more computer systems. The logical operations of the present invention are implemented (1) as a sequence of processor-implemented steps executing in one or more computer systems and (2) as interconnected machine modules within one or more computer systems. The implementation is a matter of choice, dependent on the performance requirements of the computer system implementing the invention. Accordingly, the logical operations making up the embodiments of the invention described herein are referred to variously as operations, steps, objects, or modules.  
         [0077]     The above specification, examples and data provide a complete description of the structure and use of exemplary embodiments of the invention. Since many embodiments of the invention can be made without departing from the spirit and scope of the invention, the invention resides in the claims hereinafter appended.