### Table 3: Classification error obtained on USPS with SVM, Parzen windows and Manifold Parzen windows classifiers.

2003

"... In PAGE 6: ... The original training set (7291) was split into a training (first 6291) and validation set (last 1000), used to tune hyper-parameters. The classification errors for all three methods are compared in Table3 , where the hyper-parameters are chosen based on validation classifi- cation error. The log-likelihoods are compared in Table 4, where the hyper-parameters are chosen based on validation ANCLL.... ..."

Cited by 14

### Table 3: Classification error obtained on USPS with SVM, Parzen windows and Manifold Parzen windows classifiers.

2003

"... In PAGE 7: ... The original training set (7291) was split into a training (first 6291) and validation set (last 1000), used to tune hyper-parameters. The classification errors for all three methods are compared in Table3 , where the hyper-parameters are chosen based on validation classifi- cation error. The log-likelihoods are compared in Table 4, where the hyper-parameters are chosen based on validation ANCLL.... ..."

Cited by 14

### Table 3: Classification error obtained on USPS with SVM, Parzen windows and Manifold Parzen windows classifiers.

"... In PAGE 7: ... The original training set (7291) was split into a training (first 6291) and validation set (last 1000), used to tune hyper-parameters. The classification errors for all three methods are compared in Table3 , where the hyper-parameters are chosen based on validation classifi- cation error. The log-likelihoods are compared in Table 4, where the hyper-parameters are chosen based on validation ANCLL.... ..."

### Table 1: Correspondence between singularities of tangents of the manifold, the 2-parameter family of height func- tions, and the pedal surface. There are two singularities of co-dimension one: curves of cusps and curves of self- intersections (xings). There are six singularities of co-

"... In PAGE 5: ... In each case, the singular- ity is defined by two pairs of critical points and we get two types each because these pairs may be disjoint or share one of the points. See Table1 for the features on a0 that corre- spond to the six types of co-dimension two singularities. We can now be more precise about what we mean by a generic 2-manifold.... In PAGE 5: ... GENERICITY ASSUMPTION A. The 2-parameter family of height functions on a0 has no violations of Conditions I and II for Morse functions other than the ones men- tioned above (and enumerated in Table1 below). Some of these violations will be discussed in more detail later as they can be locations of maximum elevation.... In PAGE 6: ... The only re- maining possibility for discontinuous elevation is thus at in- terchanges, which happen when two points share the same tangent plane. As mentioned in Table1 , this corresponds to a point at which the pedal surface intersects itself. Figure 6 shows that discontinuities in the elevation can indeed arise at co-tangent points.... In PAGE 8: ... We know that a7 is not a flat point of a0 , else its elevation would be zero. This simple observation eliminates five of the eight singularities in Table1 . Furthermore, the assumption of a generic 2-manifold a0 implies that a multi- plicity three point can only be paired with a multiplicity one point.... ..."

### Table 1: Correspondence between singularities of tangents of the manifold, the 2-parameter family of height functions, and the pedal surface. There are two singularities of co-dimension one: curves of cusps and curves of self-intersections (xings). There are six singu- larities of co-dimension two.

"... In PAGE 4: ... In each case, the singularity is defined by two pairs of critical points and we get two types each because these pairs may be disjoint or share one of the points. See Table1 for the features on C5 that correspond to the six types of co-dimension two singularities. We can now be more precise about what we mean by a generic 2-manifold.... In PAGE 4: ... GENERICITY ASSUMPTION A. The 2-parameter family of height functions on C5 has no violations of Conditions I and II for Morse functions other than the ones mentioned above (and enumerated in Table1 below). Some of these violations will be discussed in more detail later as they can be locations of maximum elevation.... In PAGE 5: ... The only remaining possibility for discontinuous elevation is thus at interchanges, which happen when two points share the same tangent plane. As mentioned in Table1 , this corresponds to a point at which the pedal surface intersects itself. Figure 6 shows that discontinuities in the elevation can indeed arise at co-tangent points.... In PAGE 6: ... We know that DC is not a flat point of C5 , else its elevation would be zero. This simple observation eliminates five of the eight singularities in Table1 . Furthermore, the assumption of a generic 2-manifold C5 implies that a multiplicity three point can only be paired with a multiplicity one point.... ..."

### Table 4. Results of min-max and hyperbolic tangent score normalization methods.

"... In PAGE 5: ... We used the sum rule for classifier combination. The resulting false identification rates are shown in Table4 . As can be observed, there is no significant performance difference between using min-max or hyperbolic tangent methods for score normalization.... ..."

### Table 2. Classification error rates (%) and standard deviations when GLKS-SSKM is compared with GLKS-SSL. Euclidean Transformation Tangent All

2007

"... In PAGE 7: ..., 2006), we run GLKS-SSKM and GLKS-SSL on four different convex sets of graph Laplacian kernels, with three sets based on the three distance metrics and the last one based on all col- lected metrics. Table2 summarizes the experimen- tal results. GLKS-SSKM outperforms GLKS-SSL in different cases, showing that integrating the cluster assumption and the manifold assumption does help.... ..."

Cited by 1

### Table 1. Intrusion Detection Results Category Number Used Classified as Human Classified as Animal Classified as Vehicle

"... In PAGE 8: ... Performance was evaluated using precision and recall method. Table1 shows the summary of test result. Matching in tangent space has the highest human intrusion detection precision of 96.... ..."

### Table 3: Classification error obtained on USPS with SVM, Parzen windows and Manifold Parzen

2003

"... In PAGE 7: ... The original training set (7291) was split into a training (first 6291) and validation set (last 1000), used to tune hyper-parameters. The classification errors for all three methods are compared in Table3 , where the hyper-parameters are chosen based on validation classifi- cation error. The log-likelihoods are compared in Table 4, where the hyper-parameters are chosen based on validation ANCLL.... ..."

Cited by 14

### Table 3: Classification error obtained on USPS with SVM, Parzen windows and Manifold Parzen

2003

Cited by 14