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M-space embedding

Training objective: encoder \(f : X \to z\) preserves the multi-scale spectral clustering structure of the input features. Same method for decoder omitted.

Notation

  • \(X \in \mathbb{R}^{N \times D}\) : input features, \(\quad z = f(X) \in \mathbb{R}^{N \times d} \quad (d \ll D)\)
  • \(W \in \mathbb{R}^{N \times N}\) : RBF affinity, \(\quad W_{ij} = \exp(-\|x_i - x_j\|^2 / 2\sigma^2), \quad \sigma\) adaptively chosen by mean degree
  • \(A = SWS\) : symmetric normalization, \(\quad S = \operatorname{diag}(D^{-1/2}), \quad D = \operatorname{diag}(W1)\)
  • \(A = U \Lambda U^T\) : eigenvalues descending, \(\quad U_k = U[:, :k]\) (top-\(k\) eigenvectors)
  • \(P_k = U_k U_k^T \in \mathbb{R}^{N \times N}\) : projector onto the top-\(k\) eigenspace

Flag space \(P\) stack

For a set of scales \(n\_eig\_list = [k_1, \ldots, k_B]\), computed once from \(X\):

\[ P_{\mathrm{gt}} = \operatorname{stack}_b(P_{k_b}) \in \mathbb{R}^{B \times N \times N} \]

Each \(k_b\) is one clustering granularity (small \(k \to\) coarse, large \(k \to\) fine).

Training Loss

We want to train encoder \(f : X \to z\), train the \(f\).

Loss typeA: z_eigvec

Repeat the full pipeline on \(z\) to get \(\hat A, \hat U, \hat \lambda\) and projectors \(\hat P_{k_b} = \hat U_{k_b}\hat U_{k_b}^{T}\).

\[ \mathcal{L}^{z\text{-eigvec}}_{\mathrm{flag}} = \frac{1}{BN^2} \sum_{b=1}^{B} \left\| P_{k_b} - \hat P_{k_b} \right\|_F^2 \]
  • Per-scale alignment: each \(k\) requires the top-\(k\) eigenspace of \(z\) to match that of the input.
  • Backward uses the \(H/(\lambda_S - \lambda_\perp)\) formula, stable under eigenvalue degeneracy.

Loss typeB: z

No spectral decomposition; use embedding inner products directly:

\[ \hat P = zz^T \in \mathbb{R}^{N \times N}, \qquad \mathcal{L}^{z}_{\mathrm{flag}} = \left\| \frac{1}{B}\sum_{b}^{B} P_{k_b} - zz^T \right\|_F^2 \]
\[ \bar P = \frac{1}{B}\sum_b P_{k_b} \quad \text{(single matrix).} \]
  • Target collapses to the "mean" of multi-scale projectors.