### Table 1 Properties of techniques for dimensionality reduction.

"... In PAGE 11: ...2. General properties In Table1 , the thirteen dimensionality reduction tech- niques are listed by four general properties: (1) the con- vexity of the optimization problem, (2) the main free... In PAGE 11: ... We discuss the four general properties below. For property 1, Table1 shows that most techniques for dimensionality reduction optimize a convex cost func- tion. This is advantageous, because it allows for find- ing the global optimum of the cost function.... In PAGE 11: ... Because of their nonconvex cost functions, autoencoders, LLC, and manifold charting may suffer from getting stuck in local optima. For property 2, Table1 shows that most nonlinear tech- niques for dimensionality reduction all have free param- eters that need to be optimized. By free parameters, we mean parameters that directly influence the cost func- tion that is optimized.... In PAGE 11: ... The main advantage of the presence of free parameters is that they provide more flexibility to the technique, whereas their main disadvantage is that they need to be tuned to optimize the performance of the di- mensionality reduction technique. For properties 3 and 4, Table1 provides insight into the computational and memory complexities of the com- putationally most expensive algorithmic components of the techniques. The computational complexity of a di- mensionality reduction technique is of importance to its applicability.... In PAGE 12: ...duction technique is determined by data properties such as the number of datapoints n, the original dimension- ality D, the target dimensionality d, and by parameters of the techniques, such as the number of nearest neigh- bors k (for techniques based on neighborhood graphs) and the number of iterations i (for iterative techniques). In Table1 , p denotes the ratio of nonzero elements in a sparse matrix to the total number of elements, m indi- cates the number of local models in a mixture of factor analyzers, and w is the number of weights in a neural network. Below, we discuss the computational complex- ity and the memory complexity of each of the entries in the table.... ..."

### Table 1. Error rates for classification using pre-processing dimensionality reduction versus full dimensional data

2005

"... In PAGE 3: ...et. In all the experiments a 3-NN classifier was used. We tested 50 sample points per training set and repeated for 20 random training sets. Table1 shows the average error rates as a function of the number of training samples. As it can be seen, the CCDR algorithm outperforms the other methods.... ..."

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### Table 4. Predictive error (%) of classification algorithms, using a hybrid dimensionality reduction scheme ((ii) from section 1)

"... In PAGE 9: ... We processed data on each of seven data sets using the above scheme, then we re-ran the experiments. These ex- perimental results show that, in going from the PLS scheme in Table 3 to the hybrid scheme in Table4 , there are significant across the board increases in clas- sification accuracy. In some data set like the lung cancer data set, the predictive accuracies were extremely high (something like 99.... ..."

### Table 2: Field multiplication times (in s) of our implementations for F2m on an 800 MHz Intel Pentium III. Input and output are in normal basis representation for the five rightmost columns. The compilers are GNU C 2.95 (gcc) and Intel 6.0 (icc) on Linux (kernel 2.4).

2006

"... In PAGE 13: ... Wu et al. [33, Table2 ] give sample minima (for several m 2T153; 235U) for the number of consecutive coefficients of an R-element that will permit recovery of the associated field element. Experimentally, times for Algorithm 9 for m D 163 on an Intel Pentium III are a factor 7 slower than field multiplication for a polynomial basis representation.... In PAGE 14: ... The implementation here has received limited such tuning for gcc. Table2 shows the running times from our implementation. The fastest times show that Algorithm 7 is 13% to 29% faster than the other direct multiplication algorithms for the entries with T 4, and competitive for T D 2.... In PAGE 15: ... For point operations involving only field addition, multiplication, and squaring, a polynomial-based squaring operation is sufficiently fast relative to multiplication that the squarings are typically ignored in rough estimates of point operation cost. The times in Table2 are significantly faster than in earlier papers, and suggest (at least on this platform) that multiplication for Gaussian normal bases is much closer in performance to multiplication in a poly- nomial basis than previously believed. While the difference is still sufficiently large to discourage the use of normal bases for traditional elliptic curve point operations of addition and doubling, we consider the implications for Koblitz curves and methods based on point halving.... In PAGE 16: ... Point addition requires 8 multiplications (assuming mixed coordinates). Regardless of method (basis conversion, direct, or ring mapping), Table2 suggests that the added costs of normal basis multiplication in point addition will overwhelm the relatively small savings in squarings. Point halving Halving-based methods [17, 28] replace most point doubles by a potentially faster halving operation.... ..."

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### Table 1: Advantages of aggressive dimensionality reduction

2001

"... In PAGE 10: ... The rationale behind these methods is that any change in the nearest neighbor from the full dimensionality leads to loss of information; the rationale behind our approachistobe aggressive in removing the dimensions whichhavelow co- herence as noise; thus, on an overall basis the aggressiveness of a dimensionality reduction process which uses the coher- ence probability of the dimensions may lead to very low precision with respect to the original data but much higher e#0Bectiveness and coherence. In order to illustrate our point, wehave indicated #28in Table1 #29 the prediction accuracy us- ing a 1#25-thresholding technique in which only those eigen- values which are less than 1#25 of the largest eigenvalue are discarded. This prediction accuracy is typically very close to the full dimensional accuracy and is signi#0Ccantly lower than the optimal accuracy for all 3 data sets #28as illustrated in the accuracy charts of Figures 5, 8, 11#29.... In PAGE 10: ... Thus, such a drastic reduction in dimensionality does not attempt to mirror the original nearest neighbors in the data; but rather improves their qualityby removing the noise e#0Bects in high dimensionality. It is also clear from Table1 that the opti- mal accuracy dimensionality is signi#0Ccantly lower than the 1#25-thresholding method. In fact, the dimensionality for the 1#25-thresholding method is quite close to the full dimension- ality.... ..."

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### Table1. Key requirements on instrument parameters for chemiluminescence, electrochemical sensor NO gas analysis of existing and novel type, for use in exhaled breath measurement in clinical settings

"... In PAGE 4: ... In order to provide an instrument for use outside the specified labora- tory environment, the aim was an instrument, with a mass reduction at a factor of 40 and price by at least a factor of 10, compared to existing devices. More specifically, the aims are listed in Table1 , together with corresponding data of present devices and including existing versions of one of the candidate new sensor technologies. POTENTIAL NEW SOLUTIONS The potentially applicable sensor technologies were scanned and after an initial pre-study, the electrochemical sensor technology was chosen for further adaptation into a final instrument concept.... ..."

Cited by 1

### Table 3 Dimensionality Reduction Procedure (Original Data) Attribute Mutual Conditional Conditional Iteration Name Information MI Entropy

"... In PAGE 13: ... Only three out of six candidate input attributes (Age, Month, Gender, Region, Religion, and Continent of Birth) have been identified as significant and included in the Information-Theoretic Network. The selected input attributes and the degree of their association with the target attribute (Diagnosis) are represented in Table3 below. Table 3 Dimensionality Reduction Procedure (Original Data) Attribute Mutual Conditional Conditional Iteration Name Information MI Entropy ... In PAGE 13: ... Table3 shows three information-theoretic measures of association between the input attributes and the target attribute: Mutual Information, Conditional Mutual Information, and Conditional Entropy. All these parameters are based on the notion of Entropy (see Cover, 1991), which represents the uncertainty degree of a random variable.... In PAGE 14: ....097 only vs. 0.127 with the original data. Thus, we have decided to keep the first model (represented in Table3 ) for the subsequent stages of the ... In PAGE 17: ... Both the rules having the highest positive and the lowest negative weights are of a potential interest to a user. Due to the disjunctive structure of the information-theoretic network, the sum of connection weights assigned to all the rules is equal to the estimated mutual information between a set of input attributes and a target attribute (see Table3 above). The information-theoretic network constructed from the original data (see sub-section 5.... ..."

### Table 6: Confusion matrix analysis on training data. (a)

"... In PAGE 4: ... Based on this merit, the engine test model is performing a classification task; it classifies the tested engine oil to be in one of the following 4 viscosity-increase ranges: Gb7 lt; 100 Gb7 between 100 and 200 Gb7 between 200 and 375 Gb7 gt; 375 We analyze the accuracy of the GP model in classifying the engine test data using confusion matrices. In Table6 and 7, each row represents the actual values while the column gives the predicted value. As shown, the model is very good at predicting lt; 100 range.... ..."

### Table 1: A comparison of timing results for the original and modi ed Symmetric- Galerkin boundary integral algorithm for the two dimensional Laplace equation. Shown are the times for computing the nonsingular integrals in Eq. (4), and for the complete solution algorithm.

"... In PAGE 7: ... (5) are exam- ined separately by solving a Dirichlet and a Neumann problem. Table1 presents computation times, for the Laplace equation, for the Dirichlet problem, Table 2 dis- plays the analogous results for a Neumann problem. Results for three discretizations, M = 150; 300; 400, M being the number of nodes, are reported.... In PAGE 8: ...80 20.1 Table 2: Timing results, as in Table1 , for Eq. (5).... ..."

### Table 1 Numbers of Bi-CGSTAB iterations required Laplace operator matrix, sherman5, and BFS for eight di erent summation orderings of the dot products. Data from Etsuko Mizukami.

75

"... In PAGE 3: ... Finally, each domain decomposition implicitly de nes a reordering of the matrix with subsequent changes in the order of operations and quality of preconditioning. Table1 shows the number of iterations required by Bi-CGSTAB for the matrix sherman5 from the Harwell-Boeing collection of test matrices, a steady-state backward-facing step problem in CFD, and the Laplace operator on a 24 24 24 cube discretized using centered di erences. Only the order of summation used in computing the dense dot products was varied; the partitioning and other computations were kept xed.... In PAGE 3: ... Even this simple change causes the number of iterations to vary by over 20%. Although this seems an unusual result which indicates an ill-conditioned system or unstable algorithm, it commonly occurs even with well-conditioned problems: the three in Table1 have estimated condition numbers of 4:2 102, 3:6 105, and 1:3 104. CG- like iterative methods rarely have monotonic convergence with respect to the residual norm, and even CG applied to symmetric positive de nite systems characteristically has sharp drops followed by plateaus.... ..."