Sketch Tomography: Inverse Problem Techniques

• Sketch tomography is a family of techniques that uses compressed data representations, like binary silhouettes and observable sketches, to solve inverse problems.
• In imaging, multiresolution sketching leverages block-averaging and primal-dual optimization to achieve efficient, linearly convergent reconstructions.
• In quantum state estimation, hybrid methods combine classical shadow measurements with low-rank tensor models to outperform traditional approaches in sample efficiency.

Sketch tomography refers to a family of techniques for solving inverse problems, particularly in imaging and quantum state estimation, in which low-dimensional or compressed "sketches" of data or the object are used to constrain or guide the reconstruction. The term encompasses both classical and quantum variants, including the use of binary silhouettes in computed tomography (CT), multiresolution image sketches in large-scale imaging, and hybrid shadow protocols for quantum state tomography. Across these modalities, sketch tomography substitutes or augments full measurement data with succinct, often highly structured summaries, leading to alternative inverse formulations, reduced computational requirements, and new theoretical considerations (Bell et al., 11 Feb 2024, Perelli et al., 13 Dec 2024, Tang et al., 3 Dec 2025).

1. Sketch Tomography in Discrete Tomographic Imaging

One central instance is silhouette tomography, a subcase of sketch tomography in which each measurement—corresponding to a ray in CT—returns a single bit encoding whether any part of the object is intersected (the silhouette). Mathematically, the object is modeled as a binary density $f\in\{0,1\}^N$ and the set of observed silhouettes as $s\in\{0,1\}^M$. The forward operator is $s = S(f) := T_{>0}(Hf)$, where $H\in\mathbb{R}^{M\times N}$ models the line integrals and $T_{>0}$ applies pointwise thresholding. The inverse problem is to reconstruct $f$ given $s$, i.e., find $f\in\{0,1\}^N$ such that $T_{>0}(Hf)=s$.

This highly non-invertible mapping admits multiple solutions. The binary feasible set is characterized by necessary constraints: for each measurement ray, if $s_m=1$ then at least one intersecting voxel must be nonzero, and if $s_m=0$ all intersected voxels must vanish.

A closed-form maximal-norm solution exists: $f_{\mathrm{max}}(s) = 1 - T_{>0}(H^T(1-s))$ which backprojects the zero-silhouettes, thresholds, and then negates, yielding the sparsest solution consistent with the measurements (Bell et al., 11 Feb 2024).

2. Multiresolution Image Sketching and Optimization

For high-dimensional imaging tasks, sketch tomography generalizes to the use of multiresolution image sketching, as in the ImaSk method for large-scale CT (Perelli et al., 13 Dec 2024). Here, sketches are created by applying block-averaging operators $T_i$ to image vectors $x\in\mathbb{R}^d$, producing lower-dimensional representations at multiple resolutions. The composition $S_i = T_i^* T_i$ provides a symmetrized sketch operator, and the system of sketches $\{S_i\}$ satisfies an unbiasedness condition $\sum_i p_i S_i = I$ for chosen probabilities $\{p_i\}$.

In the primal-dual saddle-point reformulation of inverse problems,

$$\min_x \max_y \sum_i \langle y, K S_i x \rangle - F^*(y) + R(x)$$

stochastic updating using randomly selected sketches $S_{i_k}$ leads to high-efficiency iterative solvers. Notably, the convergence theorem guarantees linear convergence provided strong convexity in the regularization $R(x)$ and unbiased sketching, with per-iteration cost far below full-resolution projections (Perelli et al., 13 Dec 2024).

3. Sketch Tomography in Quantum State Estimation

In quantum tomography, sketch tomography hybridizes the classical shadow protocol with matrix product state (MPS) parameterizations (Tang et al., 3 Dec 2025). For quantum states admitting an efficient MPS or tensor-train (TT) representation,

$$\rho = \sum_{i_1,\dots,i_n=1}^4 C(i_1,\dots,i_n) \,\sigma_1^{i_1}\otimes\cdots\otimes\sigma_n^{i_n}$$

with $C$ decomposed into low-rank TT cores $\{G_k\}$, the algorithm reconstructs $\rho$ by estimating TT cores directly from observable sketches using a combination of Pauli-shadow measurements and least-squares fitting on core sketch matrices $B_k$.

The procedure provably converges with sample complexity $O(n^2/\epsilon^2)$ for $n$-qubit systems, outperforming both naive classical shadow and maximum likelihood estimation (MLE) within the MPS class on several benchmark tasks. It is particularly effective for global observables and entropic quantities in area-law or quasi-1D quantum systems (Tang et al., 3 Dec 2025).

4. Data, Architecture, and Implementation Details

The implementation of sketch tomography varies by modality:

• For silhouette tomography in classical CT, training data consists of binary volumetric shapes and their associated silhouette observations, often generated synthetically from 3D model databases such as ShapeNet. Deep neural networks, typically U-Nets, are trained on backprojected silhouette data using mean-squared error loss, yielding occupancy probability maps or thresholded binary reconstructions (Bell et al., 11 Feb 2024).
• In multiresolution sketching, the ImaSk algorithm leverages block-averaged sketches at varying resolutions and maintains memory tables for efficient SAGA-style primal-dual updates, with step-size and probability parameters tuned for linear convergence according to the theory (Perelli et al., 13 Dec 2024).
• In quantum sketch tomography, classical shadow measurements are collected and processed into sketches of observable statistics, from which the TT/MPS parameters are estimated via SVD and least-squares methods (Tang et al., 3 Dec 2025).

Table: Comparison of Sketch Tomography Paradigms

Modality	Forward Operator	Sketch Type
Discrete CT	$T_{>0}(Hf)$	Silhouette (binary)
Large-scale CT	$K S_i x$	Multiresolution
Quantum	Pauli-shadow + TT	Observable sketches

5. Experimental Results and Benchmarks

• In silhouette tomography evaluated on synthetic datasets from ShapeNet, U-Net-based reconstructions achieve MSE $0.0229$ and SSIM $0.875$ (or $0.952$ after thresholding) versus a maximal solution baseline with MSE $0.0628$, indicating substantial accuracy improvements (Bell et al., 11 Feb 2024).
• ImaSk, operating on CT slices and thorax images, demonstrates linear convergence rates and a twofold reduction in required full-resolution forward model calls with $r=4$ or $8$ sketches, even under strong regularization such as total variation.
• Quantum sketch tomography, tested on Heisenberg and transverse-field Ising models (up to $n=64$ qubits), shows maximum Renyi entropy errors below $0.01$ and outperforms classical shadow and MLE-based MPS on difficult global and entropic observables (Tang et al., 3 Dec 2025).

6. Extensions, Limitations, and Theoretical Considerations

Sketch tomography encompasses a range of possible sketch operators and network architectures:

• The forward operator $S$ may generalize from strict silhouettes to signed distance maps, masked line integrals, or occupancy probabilities along user-selected 2D curves.
• Expansion to alternative tensor networks (e.g., PEPS, tree-tensor, hierarchical Tucker) and shadow bases is possible in the quantum setting, subject to analogous sketching principles (Tang et al., 3 Dec 2025).
• Inverse ill-posedness can be mitigated by richer or more informative sketch constraints, at the expense of higher model complexity or training data requirements.
• A major limitation is the requirement of a suitable model class: binary volumes in discrete CT, moderate-rank MPS for quantum systems, or sufficient training data for learned models.
• Theoretical results guarantee geometric convergence under strong convexity and unbiased sketching (image domain), and near-optimal sample efficiency (quantum states), but worst-case hardness is not alleviated in the fully general case (Bell et al., 11 Feb 2024, Perelli et al., 13 Dec 2024, Tang et al., 3 Dec 2025).

In summary, sketch tomography unifies a set of powerful techniques for inference from highly compressed or structured measurements across imaging and quantum domains, leveraging bespoke sketch operators, advanced optimization algorithms, and, where appropriate, supervised deep learning. The methodology yields theoretical and empirical gains in efficiency and accuracy, particularly when the underlying model class aligns with the sketch format.