Summary of non-negative matrix factorization (NMF) applied to odor profiling data.

<p> Schematic Overview: NMF seeks a lower, <i>s</i>-dimensional approximation of a matrix as the product of matrices and . is , consisting in the present study of odor descriptors odors. A given column of is the semantic profile of one odor, with each entry providing the percent-used value (see methods) of a given descriptor. Columns of are basis vectors of the reduced, s-dimensional odor descriptor space. Columns of are -dimensional representations (weights) of the odors in the new basis. Plot of residual error between perceptual data, , and different NMF-derived approximations. . For each choice of subspace, data were divided into random training and testing halves, and residual error between and computed. One-hundred such divisions into training and testing were used to compute the standard errors shown (shaded areas). Reconstruction error (fraction of <i>unexplained</i> variance) for PCA and NMF vs. number of dimensions. The change in reconstruction error for the first interval is indicated by asterisks(*), and corresponds to the first point in the next panel. Change in reconstruction error for PCA and NMF, compared to the change in reconstruction error for PCA performed on a scrambled matrix (). is used to estimate the cutoff number of dimensions for which a given dimensionality reduction method is explaining only noise in a dataset. Note that each point, , is actually the difference in reconstruction error between dimensions and (by way of illustration, points with an asterisk in this panel denote corresponding intervals in the previous panel ).