spectralbrain.compute_bates_signatures#
- spectralbrain.compute_bates_signatures(decomp, t_values=None, *, n_times=10, order=2)[source]#
Symmetric polynomial signatures — sign/ordering invariant.
Construct weighted eigenfunctions w_j(x, t) = exp(-λ_j·t)·φ_j(x), then compute elementary symmetric polynomials e_p of the weights. These are provably invariant under sign flips and permutations of the eigenfunctions.
\[ \begin{align}\begin{aligned}e_1(x, t) &= \sum_j w_j(x, t) \quad \text{(= HKS)}\\e_2(x, t) &= \sum_{j < k} w_j(x, t)\, w_k(x, t)\\e_p(x, t) &= \sum_{j_1 < \cdots < j_p} \prod_{m=1}^{p} w_{j_m}(x, t)\end{aligned}\end{align} \]For order=2 via Newton’s identity: e_2 = (e_1² − Σ w_j²) / 2
- Parameters:
decomp (SpectralDecomposition)
t_values (ndarray, optional) – Time scales.
None= auto.n_times (int) – Number of auto time scales.
order (int) – Maximum order of symmetric polynomials (1, 2, or 3). Higher orders are more informative but O(k^order).
- Returns:
ndarray, shape (N, order × T) – Concatenated symmetric polynomial signatures across orders and time scales.
- Return type:
References
Bates J, Pafundi D, Kanel P, Liu X, Mio W. Spectral signatures of point clouds and applications to detection of Alzheimer’s disease through neuroimaging. IEEE ISBI 2011.