Computes \(\widehat Y = X \, C \, A_{\mathrm{new}}\)
(\(= X (C_{+} - C_{-})(A_{+}^{\mathrm{new}} - A_{-}^{\mathrm{new}})\)).
For type = "response" the raw prediction is returned
(possibly signed).
With signed covariates the scores \(\bm b_n = C\bm a_n\) can be
negative, so the NMF-LAB recipe of normalizing \(\bm b_n\) to a
membership vector does not apply. Since \(X\) is column-stochastic,
\(\widehat{\bm y}_n = X\bm b_n\) still sums to \(\sum_q b_{qn}\)
(close to one for one-hot targets) and is the least-squares estimate of
the class indicator; for type = "prob" it is mapped to the
probability simplex by the Euclidean projection
$$\widehat{\bm p}_n = \mathop{\mathrm{arg\,min}}_{\bm p \ge 0,\ \bm 1^\top \bm p = 1}
\lVert \bm p - \widehat{\bm y}_n \rVert^2
= \max(\widehat{\bm y}_n - \tau_n \bm 1, 0),$$
the closest probability vector in the same Frobenius geometry the fit
minimizes (Duchi et al. 2008; Wang & Carreira-Perpinan 2013;
\(O(P \log P)\) per column by sorting). The former rule – clip
negatives to zero and renormalize – is kept as
prob.method = "clip". type = "class" is
\(\arg\max_p \widehat y_{pn}\), which both rules preserve.
Arguments
- object
A fitted
"nmfkc.signed"object.- newA
Real-valued \(D \times N_{\mathrm{new}}\) covariate matrix.
- type
Output:
"response"(raw signed),"prob", or"class".- ...
Hidden option
prob.method: howtype = "prob"maps \(\widehat{\bm y}_n\) to the simplex,"simplex"(default; Euclidean projection) or"clip"(clip negatives, renormalize).
References
Ding, C. H. Q., Li, T., & Jordan, M. I. (2010). Convex and semi-nonnegative matrix factorizations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 32(1), 45–55.
Duchi, J., Shalev-Shwartz, S., Singer, Y., & Chandra, T. (2008). Efficient projections onto the l1-ball for learning in high dimensions. Proceedings of the 25th International Conference on Machine Learning (ICML), 272–279.
Wang, W., & Carreira-Perpinan, M. A. (2013). Projection onto the probability simplex: an efficient algorithm with a simple proof, and an application. arXiv:1309.1541.