Sparse-TDA via QR Pivoting

• The paper presents a sparse TDA approach that leverages pivoted QR factorization to select informative pixels and reduce feature dimensions efficiently.
• It combines persistent homology, persistence images, and truncated SVD to convert complex topological data into actionable features for machine learning.
• Empirical results show that Sparse-TDA significantly lowers computation time while maintaining competitive accuracy compared to kernel-based TDA methods.

Sparse-TDA via QR Pivoting is a method for sparse realization of topological data analysis (TDA) in supervised multi-way classification settings. It combines persistent topological feature extraction with principled column selection via pivoted QR factorization, enabling efficient, information-preserving dimension reduction for subsequent machine learning tasks. Key components include the vectorization of persistence diagrams (PDs) into persistence images (PIs), low-rank approximation via truncated SVD, and sparse pixel selection using QR pivoting over the PI domain. Benchmark evaluations indicate that Sparse-TDA maintains competitive accuracy with significant computational savings compared to kernel-based TDA approaches (Guo et al., 2017).

1. Persistent Topological Feature Vectorization

Persistent homology summarizes topological features of data through PDs, representing birth–death pairs $(x,y)$ corresponding to the lifetime of topological structures. To enable statistical learning, these diagrams are transformed into vector-valued features:

• Linear Coordinate Transformation: Each PD $D_i\subset\R^2$ undergoes the map $T(x,y) = (x,\, y-x)$, so coordinates represent birth time and persistence.
• Persistence Surfaces: Define

$\rho_{D_i}(z) = \sum_{u\in T(D_i)} f(u)\,g_u(z),$

where $f(u)$ is a weighting (e.g., linear $u_y/u_y^*$ or nonlinear $\arctan(c\,u_y)$), and $g_u(z)$ is a Gaussian with variance $\sigma^2$ centered at $u$.

• Persistence Images (PI): $\rho_{D_i}$ is discretized on a $p \times p$ grid; each cell’s integral forms a component of the vector $\mathbf{x}_i \in \R^{d}$ with $d = p^2$.
• Data Matrix Assembly: The PI representations for $n$ samples form $X \in \R^{n\times d}$.

This process yields a standardized, high-dimensional encoding of persistent topological information suitable for downstream processing.

2. Low-Rank Approximation via SVD

Due to the invariance and redundancy in the PI representations, the data matrix $X$ admits a low-rank structure:

• Economy-Size SVD: Compute $X \approx U_r \Sigma_r V_r^T$, where $U_r \in \R^{n \times r}$, $V_r \in \R^{d \times r}$, and $\Sigma_r$ holds the top-r singular values.
• Rank Selection: The optimal $r$ is determined by the Gavish–Donoho rule: retain all $\sigma_j$ exceeding $4\sqrt{\beta}$ times the median ($\beta = n/d$), providing a near-minimax approximation in the presence of Gaussian noise.

This step isolates the dominant variance directions, yielding a subspace with maximal topological signal representation.

3. QR Pivoting for Sparse Pixel Selection

Sparse-TDA achieves drastic reduction in feature dimensionality via column subset selection:

• Pivoted QR Algorithm
  1. Input $U_r \in \R^{n \times r}$ and target sparsity $s \le r$.
  2. Compute the pivoted QR of $U_r^T$: $U_r^T P = Q R$, with $P$ the permutation matrix determined by Businger–Golub norm-maximal pivoting.
  3. The first $s$ pivots $\{\pi_1, \dots, \pi_s\}$ identify columns (pixels) most informative for the dominant subspace.
  4. For each sample, PI vectors are restricted to these $s$ elements: $\tilde{\mathbf{x}}_i = \mathbf{x}_i[\pi_1:\pi_s]$.

This mechanism provides a principled, data-driven way to select a subset of PI grid locations most critical for information retention.

4. Computational Complexity

The overall efficiency of Sparse-TDA is determined by:

Stage	Dominant Complexity	Remarks
Feature Extraction (PD→PI)	$O(n\,d\,m)$	$m$: simplices per data item
Truncated SVD	$O(n\,d\,r)$	Efficient via randomized methods
Pivoted QR	$O(r^2 n)$	Businger–Golub algorithm
SVM Training	$\approx O(n^2 s)$	Kernel method, feature dim. $s$

The SVD and QR stages dominate the pipeline. Sparse-TDA delivers computational savings primarily by reducing the SVM input dimensionality, typically setting $s=r \ll d$.

5. Multi-Way Classification Procedure

Once the sparse features are extracted, classification proceeds as follows:

• Sparse Feature Assembly: All training PIs are projected to $\tilde{X} \in \R^{n \times s}$ via QR-selected indices.
• RBF-kernel SVM Training: Use $\tilde{X}$ to train a one-versus-rest SVM; the RBF kernel is $$K(\mathbf{u},\mathbf{v}) = \exp(-\gamma\|\mathbf{u}-\mathbf{v}\|^2)$$.
• Parameter Selection: Grid search and cross-validation are performed for SVM $C$ and kernel width $\gamma$.
• Decision Function:

$$f(\tilde{\mathbf{x}}) = \arg\max_{k\in \{1, \dots, K\}} \sum_{i: y_i = k} \alpha_i K(\tilde{\mathbf{x}}_i, \tilde{\mathbf{x}}) + b_k,$$

where $\{\alpha_i, b_k\}$ are the learned dual and bias parameters.

This pipeline integrates topological feature selection and statistical classification within a unified, sparse framework.

6. Empirical Evaluation and Benchmarking

Sparse-TDA was validated on human posture and texture image classification problems:

• SHREC’14 Synthetic: 15 classes, 300 meshes; Sparse-TDA (NW) accuracy: 94.0%, SVM runtime: 25.6s.
• SHREC’14 Real: 400 meshes; Sparse-TDA (LW) accuracy: 68.8%, SVM runtime: 82.2s.
• Outex Texture: 24 classes, 480 images; Sparse-TDA (NW) accuracy: 66.0%, SVM runtime: 120s.

Key findings:

• Sparse-TDA achieves runtime reductions of one to two orders of magnitude compared to kernel-based TDA, with a modest loss in accuracy.
• Sparse-TDA consistently outperforms L1-SVM in accuracy, often with equal or better training times.

Dataset	L1-SVM Accuracy (%)	Sparse-TDA (LW)	Sparse-TDA (NW)	Kernel-TDA
SHREC Synthetic	89.6	91.5	94.0	97.8
SHREC Real	63.9	68.8	67.8	65.3
Outex Texture	55.1	62.6	66.0	69.2

Dataset	L1-SVM Time (s)	Sparse-TDA	Kernel-TDA
SHREC Synthetic	35.4	25.6	1182
SHREC Real	305	82.2	92.3
Outex Texture	106	120	5457

Performance reflects mean ± std over multiple data splits; PI grid, SVD rank $r$, and QR sample $s$ were dataset-specific ($r \approx 100-130$; $s=r$) (Guo et al., 2017).

7. Theoretical Properties and Stability

• Persistence Image Stability: The PD-to-PI transform is 1-Wasserstein stable: small input perturbations induce bounded changes in the resulting PI (Adams et al., 2017).
• Optimal SVD Truncation: The Gavish–Donoho median threshold guarantees near-optimal rank selection for signal-plus-Gaussian-noise matrices.
• QR Pivot Conditioning: The Businger–Golub strategy ensures the selected columns form a numerically well-conditioned basis. The associated interpolation error is bounded by the $(s+1)$th pivot growth factor (Drmač & Gugercin, 2015).

These theoretical properties guarantee robust, stable, and near-minimax realization of topological feature sampling, ensuring generalization and numerical reliability across diverse datasets (Guo et al., 2017).