Learned Dictionaries with Total Variation and Non-Negativity for Single-Cell Microscopy: Convergence Theory and Deterministic Multi-Channel Cell Feature Unification
This work solves the problem of robust and interpretable single-cell microscopy analysis for researchers and clinicians, though it is incremental as it builds on existing dictionary learning methods with specific regularization and constraints.
The paper tackles the problem of reconstructing single-cell microscopy signals under heterogeneous backgrounds and artifacts by introducing a variational dictionary learning algorithm with total variation regularization and non-negativity constraints, achieving convergence to the true solution at an optimal O(δ) rate under specific conditions. It also addresses multi-channel cell feature unification by proposing a deterministic joint dictionary learning framework that synthesizes a unified representation from five imaging channels, ensuring interpretability and reproducibility for clinical diagnostics.
We introduce a variational dictionary-based learning algorithm with hybrid penalization for single-cell microscopy signals. The cost functional couples a least-squares data fidelity term with total-variation (TV) regularization and a non-negativity constraint, promoting edge-preserving, physically meaningful reconstructions under heterogeneous backgrounds and imaging artifacts. We formulate the learning task with an explicit unitary (orthonormal) constraint on the dictionary operator, ensuring well-conditioned representations and predictable numerical behavior. The resulting optimization problem is solved by an alternating proximal-gradient scheme that combines smooth updates with closed-form proximal steps for non-smooth penalties. We prove that the PDHG iterates converge to the regularized minimizer under an explicit step-size condition ($ÏÏ< 1/8$), and that when the true solution satisfies a variational source condition (VSC), the regularized solution converges to the true solution at the optimal $O(δ)$ rate under a noise-proportional regularization parameter choice $λ\propto δ$. Beyond reconstruction, we address the problem of multi-channel cell feature unification: given five imaging channels of the BSCCM dataset (DPC Left, Right, Top, Bottom, and Brightfield), we propose a \emph{deterministic} approach to synthesize a unified single-cell representation. Rather than probabilistic latent encodings, we pursue a joint dictionary learning framework in which all five channels share a common dictionary, and the sparse codes across channels are combined to form a channel-agnostic cell descriptor. This deterministic unification strategy is mathematically transparent, reproducible, and directly compatible with the clinical requirement that AI systems for diagnostics must be interpretable and auditable.