Slice Projection in High Dimensions

• Slice Projection (SP) is a framework of techniques that restrict high-dimensional data to lower-dimensional slices or projections, enabling efficient visualization and analysis.
• SP methods integrate orthogonal projections, slicing hyperplanes, and Fourier transforms to address challenges in convex geometry, tomography, and data science.
• Algorithmic applications of SP range from interactive data visualization and sparse Fourier analysis to optimized polytope decomposition and region-of-interest reconstruction.

Slice Projection (SP) refers to a suite of mathematical, algorithmic, and computational techniques in which high-dimensional data, geometric objects, or functions are interrogated by restricting to, or projecting onto, lower-dimensional subspaces known as slices or projections. Applications span data visualization, Fourier analysis, convex geometry, tomography, and function spaces. SP unifies the dual operation of slicing (intersection with subspaces) and projection (mapping onto subspaces) and is foundational in the analysis and computation of high-dimensional phenomena.

1. Mathematical Foundations of Slice Projection

The general SP framework involves a high-dimensional object—typically a dataset $X\subset\mathbb{R}^p$, a signal, or a convex polytope $P\subset\mathbb{R}^d$—and a choice of a lower-dimensional subspace, often specified by an orthonormal basis $A\in\mathbb{R}^{p\times d}$ (for projections) or an affine hyperplane $$H(u,\beta)=\{x\in\mathbb{R}^d: \langle u,x\rangle=\beta\}$$ (for slices).

For data analysis and visualization:

• Projection: The $d$-dimensional projected coordinate of $x\in\mathbb{R}^p$ is $y = A^T x$ with $A^T A = I_d$.
• Slice Around a Center: Given $c\in\mathbb{R}^p$ and thickness $\epsilon>0$, the slice is $P\subset\mathbb{R}^d$0.

For convex geometry:

• Slicing: $P\subset\mathbb{R}^d$1 defines a hyperplane section.
• Projection: The orthogonal projection onto a $P\subset\mathbb{R}^d$2-dimensional subspace $P\subset\mathbb{R}^d$3, $P\subset\mathbb{R}^d$4, is $P\subset\mathbb{R}^d$5 with $P\subset\mathbb{R}^d$6 the $P\subset\mathbb{R}^d$7 matrix whose rows are $P\subset\mathbb{R}^d$8.

In Fourier analysis and tomography:

• Projection-Slice Theorem: A one-dimensional projection or slice in the time/domain corresponds to a line (or hyperplane) in the frequency domain, facilitating sparse recovery via lower-dimensional transforms.

2. Algorithmic and Computational Techniques

SP techniques are operationalized via algorithms that combine projection and slicing to efficiently interrogate structure in high dimensions:

• Interactive SP in Data Visualization: The SP viewer iterates over data $P\subset\mathbb{R}^d$9, computes projections $A\in\mathbb{R}^{p\times d}$0, computes orthogonal distances to the slice plane, and renders those lying within a given thickness. Interactive controls allow users to manipulate individual variable contributions (modifying rows of $A\in\mathbb{R}^{p\times d}$1), move the slice center $A\in\mathbb{R}^{p\times d}$2 ($A\in\mathbb{R}^{p\times d}$3), and adjust slice thickness $A\in\mathbb{R}^{p\times d}$4 (Laa et al., 2022).
• FPS-SFT (Fourier Projection-Slice Sparse Transform): For signals with $A\in\mathbb{R}^{p\times d}$5-sparse spectra, FPS-SFT iteratively samples along 1D lines (slices) parameterized by randomly chosen slopes and offsets, computes 1D FFTs, and decodes sparse frequency locations; computational complexity is $A\in\mathbb{R}^{p\times d}$6 with $A\in\mathbb{R}^{p\times d}$7 samples (Wang et al., 2017, Wang et al., 2017).
• Variable-Size Slicing in Point Clouds: For point cloud coding, SP adaptively cross-sections the 3D cloud into variable-thickness, partially overlapping slices, projects each slice to a 2D plane, and encodes geometry differentially relative to the slice base, thus optimizing for minimal self-occlusion and improved compression (Tohidi et al., 2022).
• Cell Decompositions for Polytope Slices/Projections: The space of all slices or projections of a polytope is partitioned into combinatorial chambers using hyperplane arrangements, facilitating polynomial-time optimization (for fixed dimension) of volume, face count, or polynomial integrals across all possible slices (Brandenburg et al., 2023).
• Local Fourier Slice: By representing data in polar or spherical wavelet frames, the SP operation is realized by restricting to iso-parameter sets in frequency space, giving explicit closed-form “sliced” wavelets and yielding efficient, locality-aware reconstructions (Lessig, 2018).

3. Applications Across Domains

Data Visualization and Model Interpretation

SP is central to dynamic visualization (“slice tours”) of high-dimensional datasets, allowing users to “tunnel” through the data and examine local structure or decision boundaries of classifiers. Techniques allow manual and continuous control over projections and slices, providing fine-grained sensitivity analysis with respect to variables and subspaces. This has been implemented in Mathematica (SliceDynamic) and in R’s tourr package (Laa et al., 2022).

Sparse Fourier Analysis and Tomography

The Fourier projection-slice theorem underpins efficient multidimensional sparse Fourier transforms (FPS-SFT, RFPS-SFT): multidimensional spectra are reconstructed by recovering one-dimensional projections taken along judiciously chosen slices, with robust extensions to handle noise and off-grid components via windowing and voting (Wang et al., 2017, Wang et al., 2017). In two-dimensional spectroscopy, the projection-slice theorem is exploited for lineshape parameter extraction by reducing the multidimensional fit to a collection of one-dimensional projections and slices, easing numerical and statistical analysis (Perez et al., 28 Oct 2025).

Convex and Computational Geometry

SP is fundamental in computational geometry for describing the space of all hyperplane sections (slices) and projections of convex polytopes. Combinatorial types, exact volume and face counts, and optimization of slice criteria are approached via arrangement-theoretic decompositions of the hyperplane/extrinsic parameter space (Brandenburg et al., 2023). The output-sensitive complexity of computing slices/projections is tightly linked to that of vertex enumeration, with several variants classified as VE-complete, and others as NP-hard depending on polytope representation and desired output (0804.4150).

Tomographic and Integral Geometry

On the sphere, the vertical slice transform maps a function to its integrals over spherical slices, with applications in spherical tomography and inverse problems. Explicit inversion and SVD formulas are established for reconstructing the original function, with analytic, hypersingular, and spectral methods available (Rubin, 2018). Local Fourier slice theorems enable efficient region-of-interest reconstruction in tomographic data (Lessig, 2018).

Function Spaces and Harmonic Analysis

In quaternionic function spaces, the orthogonal slice projection operator projects $A\in\mathbb{R}^{p\times d}$8 functions onto slice-regular subspaces, with explicit integral kernel descriptions and precise $A\in\mathbb{R}^{p\times d}$9 norm bounds (Arcozzi et al., 2015).

4. Theoretical Guarantees and Complexity

SP techniques afford strong theoretical performance in various contexts:

• Recovery Guarantees in Fourier SP: Under $$H(u,\beta)=\{x\in\mathbb{R}^d: \langle u,x\rangle=\beta\}$$0-sparsity, FPS-SFT achieves exact recovery with $$H(u,\beta)=\{x\in\mathbb{R}^d: \langle u,x\rangle=\beta\}$$1 samples in $$H(u,\beta)=\{x\in\mathbb{R}^d: \langle u,x\rangle=\beta\}$$2 time (Wang et al., 2017, Wang et al., 2017).
• Optimization in Polytope Slices: With dimension fixed, volumes and face counts of slices or projections are optimized in polynomial time via real-algebraic geometry methods (Brandenburg et al., 2023).
• Error Bounds in Model Reduction: In multi-slice Petrov-Galerkin contexts, enforcing that solutions lie in nested slices (thickened subspaces) yields sharper instance-optimal error bounds than classical projection, systematically leveraging prior information (Herzet et al., 2018).
• Complexity of SP: Depending on input/output format and degeneracy of projection/slice directions, complexity ranges from polynomial to VE-complete to NP-hard (0804.4150).

5. Implementation Paradigms and Software Ecosystem

SP tools have been realized in several computational ecosystems:

• Mathematica: Full-featured interactive SP workflow with graphical controls, dynamic projection/slice manipulation, and classification boundary exploration, distributed via mmtour.wl (Laa et al., 2022).
• R / tourr: Offers radial tours, manual slice tours, and linked display of projections and slices for statistical exploration (Laa et al., 2022).
• SageMath: Used for combinatorial and algebraic algorithms in polytopal slice/projection optimization and combinatorial analysis (Brandenburg et al., 2023).

6. Extensions and Generalizations

SP is generalized and contextualized in diverse mathematical settings:

• Infinite-dimensional Hilbert spaces: Multi-slice projectors extend Petrov-Galerkin methods for PDEs and model reduction (Herzet et al., 2018).
• Spherical and Geometric Tomography: The vertical slice transform on spheres, with exact SVD and inversion; connections to Radon transforms and EPD equations (Rubin, 2018).
• Wavelet-based SP: Adapts SP to localized, sparse, and scale-aware frameworks using polar/spherical wavelet dictionaries for computational savings in high-dimensional settings (Lessig, 2018).
• Quaternionic Analysis: SP as $$H(u,\beta)=\{x\in\mathbb{R}^d: \langle u,x\rangle=\beta\}$$3-projection onto slice-regular functions on quaternionic spheres with explicit kernel and $$H(u,\beta)=\{x\in\mathbb{R}^d: \langle u,x\rangle=\beta\}$$4 operator norm descriptions (Arcozzi et al., 2015).

7. Summary Table of SP Variants, Algorithms, and Complexity

SP Domain	Core Operation	Computational Guarantees
Data visualization/tour (R, Mathematica)	Orthogonal slice & projection	Interactive $$H(u,\beta)=\{x\in\mathbb{R}^d: \langle u,x\rangle=\beta\}$$5 per update
Sparse Fourier/SP theorem (FPS-SFT, RFPS-SFT)	1D DFTs on multidim slices	$$H(u,\beta)=\{x\in\mathbb{R}^d: \langle u,x\rangle=\beta\}$$6, $$H(u,\beta)=\{x\in\mathbb{R}^d: \langle u,x\rangle=\beta\}$$7 samples
Convex polytopes (polytope slicing)	Cell decomposition (arrangements)	Polynomial in $$H(u,\beta)=\{x\in\mathbb{R}^d: \langle u,x\rangle=\beta\}$$8 for fixed $$H(u,\beta)=\{x\in\mathbb{R}^d: \langle u,x\rangle=\beta\}$$9
Model reduction (multi-slice Petrov-Galerkin)	Quadratic programming w/constraints	Improved instance-optimal error bounds
Spherical tomography	Integrals over geodesic slices	Explicit inversion, SVD, stability
Harmonic analysis on quaternions	Orthogonal slice-projection	$d$0 norm bounds, explicit kernels

The Slice Projection paradigm provides a unifying theoretical framework and a diverse toolkit across mathematics, data science, signal processing, and geometry. It enables tractable, interpretable computation and analysis within otherwise intractable high-dimensional domains, grounded in robust mathematical and algorithmic principles (Laa et al., 2022, Wang et al., 2017, Tohidi et al., 2022, Lessig, 2018, Brandenburg et al., 2023, 0804.4150, Herzet et al., 2018, Rubin, 2018, Arcozzi et al., 2015, Perez et al., 28 Oct 2025).