### Table 1: Categorization of Manifold Learning Methods

2007

"... In PAGE 2: ...Table 1: Categorization of Manifold Learning Methods 2 Manifold Learning Methods and their connections to Distance Metric Learning Manifold Learning approaches can be categorized along the following two dimensions: first, the learnt embedding is linear or nonlinear; and second, the structure to be pre- served is global or local (see Table1 ). Based on the analysis in section 1, all the linear methods in Table 1 except Multidimensional Scaling (MDS), learn an explicit linear projective mapping and can be interpreted as the problem of distance metric learning.... In PAGE 2: ...Table 1: Categorization of Manifold Learning Methods 2 Manifold Learning Methods and their connections to Distance Metric Learning Manifold Learning approaches can be categorized along the following two dimensions: first, the learnt embedding is linear or nonlinear; and second, the structure to be pre- served is global or local (see Table 1). Based on the analysis in section 1, all the linear methods in Table1 except Multidimensional Scaling (MDS), learn an explicit linear projective mapping and can be interpreted as the problem of distance metric learning. MDS finds the low-rank projection that best preserves the inter-point distance matrix E.... ..."

### Table 2.1 Representations of subspace manifolds.

1998

Cited by 186

### Table 1. Invariant manifolds in the symmetric representation.

in c ○ World Scientific Publishing Company DYNAMICS OF THREE COUPLED EXCITABLE CELLS WITH D3 SYMMETRY

1998

"... In PAGE 5: ...n the following way [Ashwin et al., 1990]. A real variable which corresponds to the aver- age phase, that is: = ( 1 + 2 + 3)=3, and a complex variable = 1 + ei2 3 2 + ei4 3 3.The invariant manifolds in this coordinate system are shown in Table1 . The action of the symmetry group corresponds to (Fig.... ..."

### Table 2.2 Computational representation of subspace manifolds.

1998

Cited by 186

### Table 3: Overlap Analysis of Query Representations Ranked Retrieved Sets

"... In PAGE 4: ... Additionally, we believed that when there was great difference in the effectiveness of the single representations, QLN would outperform the other techniques. The full results of our experiments are given in Table 4 and Table 5 and the overlap analysis is given in Table3 . Examining Table 3 we see that the overlap between title and description or narrative query representations is very low, thus fusion techniques like CombMNZ are predicted to not do much better than fusion techniques like CombSUM.... ..."

### Table 1: Embedding data sets into manifolds

"... In PAGE 3: ... To evaluate the accuracy of the manifolds obtained we used several measures. Table1 compares the manifold with the best plane embedding in terms of (1) average error: the ratio of the objective function value (I) to the sum of squares of all the e ective distances, (2) av- erage expansion: the average expansion factor for pairs whose distance went up compared to the original, (3) average contraction: the average shrinking factor for pairs whose distance went down and (4) maximum dis- tortion: the product of maximum contraction (max fac- tor by which some edge length was reduced) and maxi- mum expansion (factor by which some edge length was... In PAGE 4: ... To test how well the learned manifold generalizes, we dropped at random 8% of the measured e ective dis- tances (edges) from the data sets, computed the man- ifold on the rest of the observations and made predic- tions on the 8% not used in computing the manifold. The last column in Table1 shows that the manifold prediction error is low on all the measures and is com- parable to that on the full set of values. From this we conclude that the manifold captures and generalizes wireless connectivity accurately.... ..."

### Table 2: Rankings of landmark representations.

"... In PAGE 8: ... In a post-study questionnaire subjects were asked to rank each landmark representation technique according to how easy it was to use. Table2 summarizes subject rankings for the five subjects in the pilot study. Table 2: Rankings of landmark representations.... ..."

### Table 2: Rankings of landmark representations.

1997

Cited by 18

### Table 5: Matchmaker ranking of helpers with cognitive style assigned a low weighting

"... In PAGE 9: ...If the helpee from Tables 3 and 4 had set the user model representation to apos;unimportant apos; (lowest on the five-point scale) for the weighting of cognitive style of a helper, the ranking in Table5 would have been calculated (assuming no other changes in the helpee model). A greater range of helper cognitive styles is now evident.... ..."