Progressive Homotopy Unrolling (UTOPY)

• Progressive Homotopy Unrolling (UTOPY) is a continuation-based method that gradually transforms a well-posed proxy problem into an ill-posed inverse problem using a homotopy parameter.
• It integrates neural unrolling with projected gradient descent and learned denoisers to enhance convergence, reconstruction quality, and memory efficiency.
• Experimental results on compressive sensing and deblurring demonstrate significant PSNR improvements, validating UTOPY's theoretical guarantees and practical effectiveness.

Progressive Homotopy Unrolling (UTOPY) is a continuation-based training and computational strategy that enables neural algorithm unrolling methods to transition from well-posed proxy problems to challenging ill-posed inverse problems through a sequence of blended operators and objectives. Rooted in the context of @@@@1@@@@ and polynomial system solving, UTOPY provides a principled and theoretically supported mechanism for improving convergence, reconstruction quality, and memory efficiency across a wide range of high-dimensional, ill-posed scenarios (Jacome et al., 17 Sep 2025, Breiding et al., 9 Sep 2025).

1. Mathematical Foundation: Homotopy and Inverse Problems

Let $x \in \mathbb{R}^n$ denote the ground-truth object (e.g., an image), $A \in \mathbb{R}^{m \times n}$ an ill-posed sensing operator, and $y = A x + e \in \mathbb{R}^m$ the observed measurements contaminated by noise $e$. The typical goal is to recover $x$ via:

$\hat{x} = \arg\min_{x} f(x;A) + R(x)$

where $f(x;A) = \tfrac{1}{2} \|y - A x\|_2^2$ encodes data fidelity and $R(x)$ is a regularizer (e.g., total variation, $\ell_1$ norm). Ill-posedness of $A$ can obstruct convergence for gradient-based solvers due to poor conditioning or information loss.

UTOPY introduces a homotopy parameter $\lambda \in [0,1]$ and a synthetic, well-posed operator $A_t \in \mathbb{R}^{m_t \times n}$ (e.g., $m_t > m$ or reduced blur). A convex combination path is constructed:

$$A(\lambda) = (1-\lambda) A + \lambda A_t, \qquad y(\lambda) = (1-\lambda) y + \lambda y_t$$

where $y_t = A_t x + e_t$ are synthetic measurements. As $\lambda$ moves from $1$ (well-posed) to $0$ (ill-posed), the optimization problem smoothly morphs from a tractable surrogate to the target.

For neural unrolling, a convex-combination fidelity is typically used:

$$g_\lambda(x) = (1-\lambda) \tfrac{1}{2}\|y - A x\|_2^2 + \lambda \tfrac{1}{2}\|y_t - A_t x\|_2^2$$

The homotopy schedule $\{\lambda_0, \lambda_1, \ldots, \lambda_L\}$ defines discrete levels for progressive optimization.

2. Progressive Homotopy Unrolling in Neural Algorithm Learning

In UTOPY, $T$ steps of projected gradient descent are embedded within a neural network, each parameterized by a learned denoiser $D_{\theta_t}$. At homotopy level $\lambda_k$, the $t$-th iteration is given by:

$$x^{t+1} = D_{\theta_t}\left(x^t - \eta_t \nabla_x g_{\lambda_k}(x^t)\right)$$

Optionally, Nesterov acceleration may be applied for improved convergence. Stacking $T$ such blocks yields the unrolled network:

$$x^T(\lambda_k) = U_{\Theta}(y, y_t; \lambda_k), \quad \Theta = \{\theta_1, \ldots, \theta_T, \eta_1, \ldots, \eta_T\}$$

Training iteratively sweeps $\lambda$ from $1$ to $0$ through prescribed scheduling (typically linear or exponential), enabling the model to first master an easier version of the task before confronting the true ill-posed instance.

The loss is generally a composite function (e.g., $$0.8\|\cdot\|_1 + 0.2 (1 - \mathrm{SSIM}) + 0.02 \|w \odot |F x - F\hat{x}|\|_2$$), measured only at the final $\lambda_L = 0$ level per epoch in practical implementations (Jacome et al., 17 Sep 2025).

3. Theoretical Guarantees and Properties

Under the following assumptions: (A1) $\nabla g_\lambda(x)$ is $L_\lambda$-Lipschitz and $C^1$ in $(x,\lambda)$, with $L = \sup_\lambda L_\lambda < \infty$; (A2) Each $D_\theta$ is $C^1$ and $\beta$-Lipschitz with $\beta < 1$,

and for step sizes $0 < \eta < \frac{1-\beta}{\beta L}$, the fixed-point operator

$$T_\lambda(x) = D_\theta(x - \eta \nabla g_\lambda(x))$$

is a contraction. Existence and uniqueness of a fixed point $x^\ast(\lambda) = \mathrm{fix}(T_\lambda)$ are guaranteed and the mapping $\lambda \mapsto x^\ast(\lambda)$ is $C^1$. Stability is quantified by

$$\|x^\ast(\lambda_1) - x^\ast(\lambda_2)\|_2 \leq C |\lambda_1 - \lambda_2|$$

with $C = \eta L / [1 - \beta(1 + \eta L)]$, certifying a smooth and controlled homotopy path from $\lambda=1$ to $\lambda=0$ (Jacome et al., 17 Sep 2025).

4. Homotopy Iterators: Streaming and Memory-Efficient Unrolling

Homotopy iterators generalize progressive homotopy unrolling to polynomial systems and related path-following tasks (Breiding et al., 9 Sep 2025). Given a homotopy $H(x, t)$ from a start system $G(x) = 0$ to a target system $F(x) = 0$, a homotopy iterator tracks individual solution paths $z(t)$, $t \in [0,1]$, with only one path in memory at any time.

A key structure is

struct ResultIterator solver::Solver # encodes H, step size, endgames start_iter::StartSolutionsIterator bitmask::Union{BitVector,Nothing} end

Each step advances by pulling the next start solution, applying predictor–corrector path tracking, and yielding the tracked solution—no large arrays of simultaneous paths are required. This reduces peak memory from $O(d n)$ (where $d$ is the number of paths and $n$ is dimension) to $O(n)$ plus constant overhead.

UTOPY leverages this streaming property: by decomposing a complex homotopy into subintervals $[t_i, t_{i-1}]$, each realized as a separate iterator and solver, a chain

$$I_0 \xrightarrow{f_{H_1}} I_1 \xrightarrow{f_{H_2}} \cdots \xrightarrow{f_{H_k}} I_k$$

emerges, representing the solution set at each progressive unroll stage, while maintaining strict memory efficiency and supporting early termination, composition, and lazy evaluation.

5. Experimental Demonstrations and Comparative Results

Experiments in (Jacome et al., 17 Sep 2025) employ the CelebA dataset (64x64 grayscale), addressing both compressive sensing (single-pixel imaging, $A=PW$ with $m/n=0.3$, SNR $=35$ dB) and image deblurring (Toeplitz Gaussian blur, $\sigma=5$). The UTOPY architecture is a $T=5$ block network with $D_{\theta_t}$ based on UNet$\langle32,64,128,256\rangle$.

Loss scheduling and homotopy schedules are compared:

Task	Baseline PSNR	UTOPY (linear)	UTOPY (exp)	Max Gain
Compressive sensing ($m/n$=0.3)	19.75 dB	21.19 dB (+1.44)	21.40 dB (+1.65)	2.5 dB
Deblurring ($\sigma=5 \to 1$)	21.17 dB	22.17 dB (+1.0)	–	1.0 dB

• Linear $\lambda$ schedule yields gradual improvement and slightly higher final PSNR but slower initial training gains.
• Exponential $\lambda$ schedule achieves earlier adaptation but marginally lower terminal PSNR.
• The Lipschitz continuity bound $\|\Delta x\| \leq C |\Delta \lambda|$ holds empirically (see original paper for quantitative confirmation).

The synthetic-to-target blending enables the network to "warm up" on an easier regime, thereby facilitating both higher-quality solutions and more robust convergence, particularly for highly ill-posed operators (Jacome et al., 17 Sep 2025).

6. Applications, Streaming Benefits, and Integration

Homotopy iterators, as developed in (Breiding et al., 9 Sep 2025), provide broad capabilities that directly support UTOPY's requirements:

• Decomposition into arbitrary sub-homotopies with efficient memory use.
• On-demand, lazy evaluation at each unrolled homotopy (or algorithm) stage.
• Early stopping and partial evaluation, supporting solution filtering, streaming, and functional queries (e.g., solution counts, extremal values).
• Convenient composition of multiple homotopies (e.g., as needed in complex pipelines or stages).
• Provable preservation of transversality, path-connectivity, and non-singularity on generic sub-homotopies.
• Efficient numerical realization using "push-forward" iterators for solution tracking, as provided in Julia implementations.

A notable implication is that the UTOPY paradigm, combined with homotopy iterator infrastructures, generalizes beyond neural inverse problems to a wide class of algebraic and computational inference settings, offering scalable, resource-efficient, and theoretically validated streaming unrolling mechanisms (Breiding et al., 9 Sep 2025).

7. Context, Generalizations, and Research Significance

Progressive homotopy unrolling addresses key challenges in training and deploying algorithm unrolling networks for inverse problems where the sensing operator is highly ill-conditioned or information-poor. By interpolating between a synthetic, well-posed proxy and the true target regime via a homotopy parameter, UTOPY overcomes optimization stagnation and enables smoother learning dynamics.

A plausible implication is that UTOPY, especially when coupled with streaming iterator architectures, can be applied to other domains involving difficult-to-solve optimization tasks, high-dimensional polynomial systems, or memory-constrained inference. The methodology's ability to expose intermediate solutions on-the-fly while maintaining minimal memory overhead has applications in large-scale computational algebra, solution-set compression, and real-time decision tasks.

The combination of interpretable design, theoretical guarantees, and empirical gains establishes UTOPY as a principled continuation-based approach within the broader landscape of algorithm unrolling and homotopy-based computational methods (Jacome et al., 17 Sep 2025, Breiding et al., 9 Sep 2025).