### Table 2 The characteristics of the models used in our paper. VG: Volumetric Graph; SM: Surface Mesh; NSM: Non-manifold surface mesh.

"... In PAGE 6: ... Our algorithm can achieve visually pleasing effects and runs interactively (see the accompanying video for animation demos). The statistics of the performance data and some experimental models are listed in Table 1 and Table2 . More simulating results are shown in Fig.... ..."

### Table 1: Upper and lower bounds on the number of loops in the Reeb graphs of Morse functions over orientable and non-orientable 2-manifolds with and without boundary.

### Table 1. Weightings Used for SMDP Graphs.

2007

"... In PAGE 3: ... 4. Manifold Construction Table1 presents different ways of creating the weight ma- trix W for state and state-action graphs. We explain below why these weightings were chosen.... ..."

Cited by 2

### Table 1. Weightings Used for SMDP Graphs.

2007

"... In PAGE 3: ... 4. Manifold Construction Table1 presents different ways of creating the weight ma- trix W for state and state-action graphs. We explain below why these weightings were chosen.... ..."

Cited by 2

### Table 2 Mean (and maximum) ratio between the Euclidean distance and the graph distance for n randomly drawn points, connectedwith the k-rule (k = 6) andthe SI-rule (SI =0:25)

"... In PAGE 25: ... In the case of a plane unit square, the true manifold distances are simply the Euclidean distance. Then, the distortion between the true manifold distances and the graph distances can be computedfor several value of n, as it is shown in Table2 . The addition of points does not really improve the distortion for the k-rule.... ..."

### Table 1: Manifold Variables Manifold M T @

1995

"... In PAGE 6: ... For such an observer, the velocity is just the normal vector u , and the acceleration of the normal vector is given by a = u r u = N?1h r N. A summary of the notation described above is given in Table1 of the ap- pendix. 2.... In PAGE 38: ... Foliation of M along I allows us to de ne the lapse, N, and the shift, N . The various manifolds we will consider, and some of the tensors de ned on them are summarized in Table1 . We can construct the following densities on Table 1: Manifold Variables Manifold M T @ ... ..."

Cited by 3

### Table 1. Singular configuration manifolds

2002

"... In PAGE 14: ... The other approach is more elaborate and takes a subspace of the entire con- figuration space into consideration. Observing Table1 , it can be seen that every singularity manifold shrinks to a point or a finite set of points if only the appropriate components of q are taken.... ..."

### Table 1 Residual variances given by Isomap for all examples: the size of the projectedsubset andthe way it is chosen (either by Random Selection or Vector Quantization) are indicatedin the secondandthirdcolumns; the fourth column shows the parameter k (minimal number of neighbors for the graph construction); the remaining columns show the residues for projection dimensionalities ranging from 1 to 6. The ideal dimensionality according to Isomap is given by the FFrst column of the plateau where residues hardly vary around a small value; the true intrinsic dimensionality of all manifolds is two; contrarily to eigenvalues yielded by PCA or MDS, residual variances may not decrease strictly

"... In PAGE 10: ... After selecting 1000 points and connecting each of them with their FFve nearest neighbors (k-rule, k = 5), Isomap delivers the result shown at the top of Fig. 3, with residue values given in the FFrst row of Table1 . After quantizing andconnecting with the same numbers of prototypes andneighbors, CDA yields the result shown at the bottom of Fig.... In PAGE 11: ... 6. The residual variances, in the third row of Table1 , indicate that Isomap diSOcultly compute a two-dimensional projection. According to the variances, Isomap cannot reduce the dimensionality, just because of the parasitic connection.... In PAGE 12: ... 7) has some diSOculties to FFnd a 2D representation. The residual variances, shown in Table1 conFFrm this fact: the plateau of small vari- ances starts at best at the thirdvalue. Anyway, the absence of a clear fall between the FFrst and second variances indicates that two dimensions do not suSOce to build a satisfying two-dimensional projection with Isomap, although the manifold is intrin- sically two-dimensional.... In PAGE 14: ... 9 (500 prototypes after vector quantization for both methods). Unfortunately, Isomap works diSOcultly with this non-Euclidean manifold (see the residual variances in Table1 ), mainly because it is circular. Visually, Isomap crushes the cylinder like an empty can.... In PAGE 17: ... after connecting the input data with k = 6. The eigendecomposition of the distance matrix yields the residual variances of Table1 . Since they are negligible andhardly varying starting from the thirdvalue, the three eigenvectors associatedwith the three largest eigenvalues are kept as advised in [20].... In PAGE 20: ... Contrarily to the face orientation problem, Isomap andCDA are fedwith the raw data, without preprocessing by PCA. When running Isomap without subset selection andwith parameter k=2, the residual variances are those mentioned in Table1 . Isomap does not succeed to detect that the underlying manifold is only one-dimensional, since the plateau begins only after the FFrst residual variance.... In PAGE 20: ... Isomap does not succeed to detect that the underlying manifold is only one-dimensional, since the plateau begins only after the FFrst residual variance. Indeed, when keeping two eigenvectors as indicated in Table1 andadvisedby [ 20], the 2D representation appears to be a circle. Consequently, the 1D representation is inevitably the projection of this circle on a line.... In PAGE 23: ... Isomap then computes the shortest paths and the low-dimensional coordinates. The residual variances, given in the last row of Table1 , do not decrease clearly, meaning that one dimension already suSOces to project the data set. Obviously, the second dimension is small but not negligible andFig.... ..."

### Table 2.1 Representations of subspace manifolds.

1998

Cited by 186

### 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.... ..."