100 90 80 ) (2 0 ) 10 (1 8 9 × 9 × 8 (1 6 ) 8 × 7 (1 4 ) 7 × 6 6 × 5 × 5 (1 2 ) (1 0 (8 ) 4 4 Figure 3. The classification accuracy for equal numbers of feature subspace basis vectors for CSA and PCA for the MNIST digits and Fashion-MNIST image sets. In CSA, the feature matrix Yk size shown is q # p for the case where p = q, with the p + q, basis vectors in the left (U ) and right (V ) subspaces together. For PCA, the feature vector Yk has r components (where r = p + q for each interval on the horizontal axis). Broader Perspective: Tensor-Based Information It is apparent, from the diversity of only the small sampling of applications we have mentioned for the matrix-based 2DPCA, B2DPCA, and CSA, that these methods have enhanced or contributed to solutions in disciplines other than engineering. This exemplifies the type of convergence of engineering methods with other disciplines referred to as amplification, wherein " engineering enhances or contributes to discipline X, or vice versa " in the Table 1. MNIST handwritten digits image set: classification accuracy. Accuracy (%) for a Given Number of Coefficients* × 3 × 3 2 × 2 (6 ) 50 10 60 ) CSA (MNIST Digits) PCA (MNIST Digits) CSA (Fashion-MNIST) PCA (Fashion-MNIST) 70 (4 ) Correct Match Rate (%) row (g)-(j), and for an example FashionMNIST image in the bottom row (g)-(j), for q # p = 25 # 25 , 18 # 18 , 10 # 10, and 6 # 6 (50, 36, 20, and 12 total basis eigenvectors, respectively). For comparison, the top row of Figure 4(b)-(e) shows reconstruction for the example MNIST digit image based on PCA with 50, 36, 20, and 12 eigenimages; the third row Figure 4(b)-(e) depicts reconstruction for the example Fashion-MNIST image based on PCA, also for 50, 36, 20, and 12 eigenimages. Figure 4 shows that for a given number of basis vectors, reconstruction with CSA is of higher quality. Figure 5 displays eigenvalues for CSA and first 50 for PCA (MNIST digits). Fashion-MNIST eigenvalue plots are similar (not shown). Accuracy (%) for a Given Number of Subspace Basis Vectors† context of the transdisciplinary systems engineering point of view espoused in [26]. Given the growing prevalence of information in tensor form, this should also prove true in more and more applications for related, more general, higher-order methods to which we have alluded, such as MPCA, HOOI, and HOSVD. For instance, in [27], HOSVD is used in coclustering to find coherent patterns, which are " subsets of subsets " in several examples involving multiway (or tensor) data including genomics, financial data, Table 2. Fashion-MNIST image set: classification accuracy. Accuracy (%) for a Given Number of Coefficients* Accuracy (%) for a Given Number of Subspace Basis Vectors† Coefficients* CSA PCA Basis Vectors† CSA PCA Coefficients* CSA PCA Basis Vectors† CSA PCA 4 49.59 57.90 4 49.59 57.90 4 63.01 64.89 4 63.01 64.89 9 82.48 87.65 6 82.48 77.64 9 75.91 76.09 6 75.91 70.29 16 91.94 93.22 8 91.94 85.72 16 78.19 79.52 8 78.19 73.99 25 95.15 94.91 10 95.15 88.44 25 79.93 80.81 10 79.93 75.90 36 95.85 95.51 12 95.85 90.94 36 81.49 81.55 12 81.49 78.39 49 95.58 95.47 14 95.58 93.06 49 81.97 81.79 14 81.97 78.66 64 95.42 95.11 16 95.42 93.51 64 82.42 82.06 16 82.42 79.52 81 95.23 94.94 18 95.23 94.22 81 82.74 81.76 18 82.74 79.46 100 95.03 94.81 20 95.03 94.58 100 83.05 82.02 20 83.05 80.43 *The number of coefficients for PCA is the total number of elements in the PCA feature vector (projections onto PCA principal axes); for CSA, it is the total number of elements in the q # p CSA feature matrix (pq coefficients). † The number of basis vectors is the number of retained principal components for PCA and the total number of eigenvectors in U and V subspaces for CSA (p + q). In this table, p = q. *The number of coefficients for PCA is the total number of elements in the PCA feature vector (projections onto PCA principal axes); for CSA, it is the total number of elements in the q # p CSA feature matrix (pq coefficients). † The number of basis vectors is the number of retained principal components for PCA and the total number of eigenvectors in U and V subspaces for CSA (p + q). In this table, p = q. Ja nu a r y 2021 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 31

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