The spectral properties of interpolated interaction matrices from multi-manifold learning can be used to predict emergent collective behaviors in active biological tissues.

Given a time series of single-cell (or single-particle) positional/velocity data from a living tissue, an interaction matrix M(t) can be constructed at discrete timepoints via multi-manifold learning (e.g., local tangent space alignment or diffusion maps on cell-cell feature embeddings). The claim is: (1) interpolating M(t) across time via spectral-domain interpolation (interpolating eigenvalues/eigenvectors rather than raw matrix entries) yields a continuous-time interaction operator M̂(t) whose top-k spectral gap and leading eigenvector orientation at time t predict, with correlation ρ≥0.6 (Pearson, p<0.01) and out-of-sample AUROC≥0.75, the onset and directionality of emergent collective motion (defined operationally as coherent velocity correlation length ξ exceeding 3 cell diameters, or order parameter |ψ|>0.5 in a Vicsek-style polarization measure) at time t+Δt, for Δt corresponding to 1–3 characteristic cell-cycle or migration timescales. The hypothesis is falsified if spectral-interpolation-based prediction does not outperform a naive baseline (raw-matrix linear interpolation, or lagged order-parameter autoregression) by a statistically significant margin (ΔAUROC≥0.05, bootstrap 95% CI excluding 0).

Minimum viable test: Use one publicly available live-imaging dataset of collective cell migration (e.g., MDCK monolayer wound-healing time-lapse, or zebrafish gastrulation cell-tracking data) with ≥5 independent movies, ≥200 cells each, ≥50 timepoints. Construct interaction matrices per frame via k-NN graph + local tangent space alignment (LTSA). Sparsely subsample frames (retain 50%, holding out alternating frames), spectrally interpolate to reconstruct held-out frames' matrices, and test whether spectral features (top-3 eigenvalues, leading eigenvector) predict the empirically observed order parameter and correlation length at held-out frames better than (a) raw linear interpolation of matrix entries, (b) naive autoregression of the order parameter itself.

for movie in tissue_datasets:
    tracks = load_tracking(movie)                     # (cell_id, x, y, t)
    frames = sample_frames(tracks, holdout_frac=0.5)
    M = {}
    for t in frames.sampled:
        G_t = knn_graph(frames[t], k=10)
        embed_t = LTSA(G_t, d=5)
        M[t] = affinity_matrix(embed_t)
        eigval[t], eigvec[t] = eigh(M[t])

    # Grassmann-manifold interpolation of eigenvectors (avoids sign flips)
    eigval_interp = cubic_spline_interp(eigval, frames.sampled, frames.holdout)
    eigvec_interp = grassmann_geodesic_interp(eigvec, frames.sampled, frames.holdout)
    M_hat = reconstruct(eigval_interp, eigvec_interp)

    features = extract_spectral_features(M_hat)        # gap, top-eigvec alignment, curvature
    y_true   = compute_order_parameter(tracks, frames.holdout)  # |psi|, xi

    model = LogisticRegression().fit(features_train, y_true_train)
    baseline_linear = fit_baseline(linear_interp(M), y_true)
    baseline_AR     = fit_baseline(AR_order_param(y_true_history), y_true)

    report_auroc(model, baseline_linear, baseline_AR, held_out_movie=movie)