Spectral Methods for Bioinformatics and Drug Discovery

We present a bridge framework linking spectral representation methods to core tasks in bioinformatics and drug discovery. The motivation is structural: omics and drug-response systems are high-dimensional, noisy, and data-limited in exactly the regime where low-rank spectral structure can provide compression, stability, and interpretable control coordinates.

The central representation is \[ x \approx \sum_{k=1}^{N(\varepsilon)} A_k v_k, \] where \(x\) denotes a biological state (for example expression, pathway activity, or response profile), \(v_k\) are spectral modes, and \(A_k\) are mode coefficients. The effective complexity \(N(\varepsilon)\) is task-specific and should be selected through out-of-sample error control, uncertainty calibration, and biological plausibility checks rather than by fixed dimensionality heuristics.

We map this representation to five biological surfaces: (i) denoising and subtype structure in expression data, (ii) mode-level intervention coordinates in gene regulatory networks, (iii) uncertainty-aware drug-response prediction, (iv) longitudinal memory-aware treatment dynamics, and (v) spectral geometry in molecular and target spaces. We also propose a decision-state extension \[ \mathcal{K}_t = (\Pi_t, U_t, M_t), \] where \(\Pi_t\) is the current predictive law, \(U_t\) is uncertainty, and \(M_t\) is a memory state capturing previously observed stress or resistance regimes.

This paper is intentionally programmatic. It does not claim immediate biological theorem closure or clinical deployment. Instead, it provides a rigorous transfer map, a bounded validation protocol, and explicit non-claims to prevent overreach while enabling fast empirical falsification.