SP-SPCA: Diverse PCA Methodologies

• SP-SPCA is an overloaded term representing multiple PCA adaptations, such as nonlinear principal curves, sparse penalization, robust spatial-sign analysis, Bayesian inference, and fast deflation techniques.
• Each variant employs a distinct methodology to address specific limitations of classical PCA, ranging from nonlinearity and noise robustness to computational efficiency in high-dimensional settings.
• Practitioners must disambiguate SP-SPCA based on context, as the methods vary in statistical models and optimization targets, impacting feature extraction and interpretability.

SP-SPCA is an overloaded acronym in contemporary arXiv usage rather than the name of a single canonical method. In different papers it denotes: Sequential Principal Curves Analysis, a nonlinear, unsupervised, and invertible feature extraction technique built from principal curves; Single-Parametric Sparse Principal Component Analysis, a sparse PCA formulation driven by a single equilibrium parameter; Spatial Sign based Principal Component Analysis, including a sparse SSPCA estimator based on the spatial-sign covariance; a fully Bayesian spike-and-slab sparse PCA with parameter-expanded variational inference; and, in some communities, the fast deflation method SPCA-SP based on subspace projections (Laparra et al., 2016, Hu et al., 14 Mar 2026, Feng, 2024, Ning et al., 2021, Xu et al., 2019). The acronym therefore spans nonlinear manifold coordinates, robust PCA, penalized sparse PCA, Bayesian sparse subspace estimation, and greedy projection-deflation methods.

1. Nomenclature and scope

Across the cited literature, the same acronym is attached to technically distinct objects. In the 2016 work of Laparra and collaborators, “SP-SPCA” is simply the acronym used for Sequential Principal Curves Analysis and does not designate a distinct variant. In the 2026 paper by Hu and Yang, SP-SPCA stands for Single-Parametric Sparse Principal Component Analysis. In the 2024 spatial-sign paper, SP-SPCA refers to a robust PCA framework based on the spatial-sign covariance, with a non-sparse estimator SPCA and a sparse estimator SSPCA. In the 2021 Bayesian paper, SP-SPCA means spike-and-slab Bayesian sparse PCA. In the 2019 deflation paper, the method is called SPCA-SP, though some communities may refer to it as SP-SPCA (Laparra et al., 2016, Hu et al., 14 Mar 2026, Feng, 2024, Ning et al., 2021, Xu et al., 2019).

Usage	Expansion	Core construction
SP-SPCA	Sequential Principal Curves Analysis	Curvilinear, invertible feature extraction from principal curves
SP-SPCA	Single-Parametric Sparse Principal Component Analysis	Sparse PCA with anisotropic SPPCR penalty and one equilibrium parameter
SP-SPCA	Spatial Sign based Principal Component Analysis	PCA/SSPCA on spatial-sign covariance under elliptical models
SP-SPCA	Spike-and-slab Bayesian sparse PCA	Spiked covariance model with orthogonality-aware prior and PX-CAVI
SPCA-SP / SP-SPCA	Sparse PCA via subspace projections	Fast deflation with Householder QR and post-hoc truncation

A recurrent misconception is that SP-SPCA always denotes a sparse PCA method. That is false in this corpus: Sequential Principal Curves Analysis is a nonlinear coordinate construction tied to principal curves and nonlinear ICA rather than sparse eigenvector estimation (Laparra et al., 2016).

2. Sequential Principal Curves Analysis

Sequential Principal Curves Analysis generalizes PCA by replacing straight principal components with first and secondary principal curves. The method assumes samples $x \in \mathbb{R}^d$ drawn from a smooth manifold with locally Gaussian clusters, and constructs a change of coordinates $y=\Phi(x)$ whose components are distances measured along curvilinear features under a chosen metric. Each coordinate behaves as the response of a “nonlinear sensor”: the response of sensor $k$ is obtained by projecting the sample onto curve $C_k$ and measuring distance along that curve from a reference origin $x^o$ (Laparra et al., 2016).

The geometric core is the principal-curve projection

$$u^*(x)=\arg\min_u \|x-C(u)\|_2,\qquad \pi(x)=C(u^*(x)),\qquad r(x)=x-C(u^*(x)),$$

together with the curvilinear coordinate

$s(x)=\int_0^{u^*(x)} \|C'(u)\|\,du,$

or, under a point-dependent metric, an analogous metric-induced line element. The sequential construction fits $C_1$, then recursively fits secondary curves in locally orthogonal subspaces, computes geodesic projections, measures distances $s_k(x)$, and maps them through strictly monotone transforms $F_k$ so that $y=\Phi(x)$0 (Laparra et al., 2016).

A central feature is the explicit Jacobian factorization

$y=\Phi(x)$1

where $y=\Phi(x)$2 is a concatenation of local rotations that unfold the manifold along principal curves, $y=\Phi(x)$3 is orthonormal, and $y=\Phi(x)$4 is diagonal in the local curvilinear axes. The metric can be tuned by selecting

$y=\Phi(x)$5

with $y=\Phi(x)$6 for Infomax, $y=\Phi(x)$7 for MSE minimization, and $y=\Phi(x)$8 for Euclidean behavior. The induced metric

$y=\Phi(x)$9

satisfies $k$0, linking local geometry to the data density (Laparra et al., 2016).

This framework is explicitly connected to nonlinear ICA. The stated decomposition is constructive: first unfold the manifold along principal curves, then locally equalize conditional distributions through a density-dependent metric. The entropy relation

$k$1

reduces, under the SPCA construction, to a determinant driven by $k$2 because $k$3 is orthonormal. For Infomax, choosing $k$4 as the empirical CDF of $k$5 yields uniform marginals and tends to reduce multi-information. For decorrelation, the criterion is $k$6 for $k$7, promoted by sequential conditioning and local equalization (Laparra et al., 2016).

Algorithmically, training requires the data $k$8, an origin $k$9, a dimension order, scaling constants $C_k$0, the metric exponent $C_k$1, and principal-curve drawing parameters: $C_k$2-neighborhood size, step size $C_k$3, and stiffness $C_k$4. Curve drawing uses a local-to-global procedure based on local PCA. Geodesic projection is refined iteratively from an orthogonal projection via

$C_k$5

Invertibility requires strictly monotone $C_k$6, well-defined local projections, and a sequential path that reaches the target point via geodesic projections. In local curvilinear coordinates the Jacobian is diagonal, and the inverse retraces the same sequence of curve traversals (Laparra et al., 2016).

The paper situates the method historically between the nonlinear ICA algorithm of Malo and Gutiérrez and later faster approximations such as Principal Polynomial Analysis and Dimensionality Reduction via Regression. It also notes applications to color manifolds, spatial texture patches, synthetic manifolds such as the swiss roll, domain adaptation for color constancy, and classification via manifold-dependent metrics (Laparra et al., 2016).

3. Single-Parametric Sparse Principal Component Analysis

Single-Parametric Sparse Principal Component Analysis is a sparse PCA method in which a single equilibrium parameter controls an anisotropic quadratic penalty in the PCA basis. Let $C_k$7 be centered, let $C_k$8, and write the SVD $C_k$9. The method introduces a diagonal matrix

$x^o$0

where $x^o$1 and, if $x^o$2 satisfies $x^o$3,

$x^o$4

This yields the quadratic penalty $x^o$5 with $x^o$6, so the shrinkage is direction-wise in the PCA basis rather than uniform as in ordinary ridge (Hu et al., 14 Mar 2026).

The estimator solves

$x^o$7

with $x^o$8 and $x^o$9. For fixed $$u^*(x)=\arg\min_u \|x-C(u)\|_2,\qquad \pi(x)=C(u^*(x)),\qquad r(x)=x-C(u^*(x)),$$0, the problem decouples into $$u^*(x)=\arg\min_u \|x-C(u)\|_2,\qquad \pi(x)=C(u^*(x)),\qquad r(x)=x-C(u^*(x)),$$1 SPPCSO subproblems

$$u^*(x)=\arg\min_u \|x-C(u)\|_2,\qquad \pi(x)=C(u^*(x)),\qquad r(x)=x-C(u^*(x)),$$2

Using the augmented regression

$$u^*(x)=\arg\min_u \|x-C(u)\|_2,\qquad \pi(x)=C(u^*(x)),\qquad r(x)=x-C(u^*(x)),$$3

each subproblem becomes a standard Lasso and is solved by coordinate descent. The $$u^*(x)=\arg\min_u \|x-C(u)\|_2,\qquad \pi(x)=C(u^*(x)),\qquad r(x)=x-C(u^*(x)),$$4-step is an orthogonal Procrustes update: if $$u^*(x)=\arg\min_u \|x-C(u)\|_2,\qquad \pi(x)=C(u^*(x)),\qquad r(x)=x-C(u^*(x)),$$5 has SVD $$u^*(x)=\arg\min_u \|x-C(u)\|_2,\qquad \pi(x)=C(u^*(x)),\qquad r(x)=x-C(u^*(x)),$$6, then $$u^*(x)=\arg\min_u \|x-C(u)\|_2,\qquad \pi(x)=C(u^*(x)),\qquad r(x)=x-C(u^*(x)),$$7 (Hu et al., 14 Mar 2026).

The gradient of the smooth part is

$$u^*(x)=\arg\min_u \|x-C(u)\|_2,\qquad \pi(x)=C(u^*(x)),\qquad r(x)=x-C(u^*(x)),$$8

and the coordinate update is the standard soft-thresholding rule

$$u^*(x)=\arg\min_u \|x-C(u)\|_2,\qquad \pi(x)=C(u^*(x)),\qquad r(x)=x-C(u^*(x)),$$9

with $s(x)=\int_0^{u^*(x)} \|C'(u)\|\,du,$0. The equilibrium parameter is shared across components, whereas the sparsity parameters $s(x)=\int_0^{u^*(x)} \|C'(u)\|\,du,$1 may be component-specific (Hu et al., 14 Mar 2026).

Without the $s(x)=\int_0^{u^*(x)} \|C'(u)\|\,du,$2 term, the paper’s Theorems 1–3 show that the solutions align with PCA directions: in the single-component and multi-component formulations with the SPPCR penalty alone, the estimated directions are proportional to the leading PCA loadings. The theoretical role of $s(x)=\int_0^{u^*(x)} \|C'(u)\|\,du,$3 is therefore stabilizing rather than rotational. Larger $s(x)=\int_0^{u^*(x)} \|C'(u)\|\,du,$4 values near $s(x)=\int_0^{u^*(x)} \|C'(u)\|\,du,$5 impose smaller penalties on dominant directions and favor higher explained variance; smaller feasible $s(x)=\int_0^{u^*(x)} \|C'(u)\|\,du,$6 values increase shrinkage in weaker directions and improve noise filtering (Hu et al., 14 Mar 2026).

The empirical section emphasizes variance preservation at matched or better sparsity. In a low-dimensional test, SP-SPCA achieved cumulative explained variance of approximately $s(x)=\int_0^{u^*(x)} \|C'(u)\|\,du,$7 versus approximately $s(x)=\int_0^{u^*(x)} \|C'(u)\|\,du,$8 for standard SPCA while preserving ideal block sparsity. In a high-dimensional factor setting with $s(x)=\int_0^{u^*(x)} \|C'(u)\|\,du,$9 and $C_1$0, it reached $C_1$1 cumulative variance versus $C_1$2 for SPCA. As $C_1$3 increased to $C_1$4, SP-SPCA remained between $C_1$5 and $C_1$6, whereas SPCA dropped to $C_1$7. On the crime dataset $C_1$8, to reach $C_1$9 variance SP-SPCA used $s_k(x)$0 nonzeros versus $s_k(x)$1 for SPCA; at $s_k(x)$2 and $s_k(x)$3, it used $s_k(x)$4 and $s_k(x)$5 versus $s_k(x)$6 and $s_k(x)$7. On S&P 500 returns $s_k(x)$8, improvements ranged from approximately $s_k(x)$9 to approximately $F_k$0 fewer nonzeros for the same explained variance (Hu et al., 14 Mar 2026).

The stated limitations are equally specific. If all $F_k$1 are small and close, the direction-wise penalization may not separate signal from noise; extreme collinearity still complicates variable-level selection; tuning over $F_k$2 requires care; and naive SVD or eigendecomposition can be $F_k$3 in the worst case for very large $F_k$4 (Hu et al., 14 Mar 2026).

4. Spatial-sign-based PCA and sparse SSPCA

In the spatial-sign literature, SP-SPCA denotes PCA based on the spatial-sign covariance, designed for high-dimensional data generated from elliptical models and explicitly aimed at robustness to heavy tails and outliers. For $F_k$5, the spatial sign is

$F_k$6

The population spatial-sign covariance is

$F_k$7

and the sample analogue is

$F_k$8

where $F_k$9 is the spatial median. For symmetric distributions, the influence function

$y=\Phi(x)$00

is uniformly bounded and constant in radius $y=\Phi(x)$01, which is the paper’s formal robustness argument (Feng, 2024).

Under the elliptical model

$y=\Phi(x)$02

with $y=\Phi(x)$03, the paper states that $y=\Phi(x)$04 shares the same eigenspace as the scatter matrix $y=\Phi(x)$05. If $y=\Phi(x)$06 and $y=\Phi(x)$07, then

$y=\Phi(x)$08

and the eigenvectors of $y=\Phi(x)$09 and $y=\Phi(x)$10 coincide in the same descending order. This shared eigenspace property is what makes principal-component analysis of $y=\Phi(x)$11 meaningful in place of covariance PCA (Feng, 2024).

The non-sparse estimator SPCA is simply the leading eigenvector $y=\Phi(x)$12 when $y=\Phi(x)$13 is distinct. The sparse estimator SSPCA assumes $y=\Phi(x)$14 and defines

$y=\Phi(x)$15

so that SSPCA is $y=\Phi(x)$16. The leading sparse error rate is

$y=\Phi(x)$17

up to eigengap and model-dependent constants, which the paper identifies as the optimal or minimax sparse PCA rate. The restricted-norm and Davis–Kahan-type inequalities establish nonasymptotic control for both sparse and non-sparse estimators (Feng, 2024).

Computation proceeds through a truncated power method. Starting from $y=\Phi(x)$18, one iterates $y=\Phi(x)$19, truncates to the $y=\Phi(x)$20 largest coordinates if needed, renormalizes, and stops when $y=\Phi(x)$21. Initialization may use $y=\Phi(x)$22 directly or a stronger Fantope Projection initializer. Sparsity $y=\Phi(x)$23 is selected by sample splitting with the criterion

$y=\Phi(x)$24

For multiple components the paper uses deflation in the sense of Mackey’s scheme (Feng, 2024).

A notable computational claim is that forming $y=\Phi(x)$25 costs $y=\Phi(x)$26 because it is a first-order average of outer products, whereas the Kendall’s tau estimator $y=\Phi(x)$27 is a second-order U-statistic with $y=\Phi(x)$28 cost. The empirical section reports that SSPCA is more accurate than ECA and TP under heavy tails, while under Gaussian data SSPCA, ECA, and TP perform similarly. On S&P 500 monthly returns from 2005–2018 $y=\Phi(x)$29, using a leverage threshold of $y=\Phi(x)$30, TP flagged $y=\Phi(x)$31 high-leverage points, ECA flagged $y=\Phi(x)$32, and SSPCA flagged only $y=\Phi(x)$33, which the paper interprets as greater robustness to crisis-period observations (Feng, 2024).

The framework’s assumptions are explicit: elliptical data, a distinct eigengap, $y=\Phi(x)$34 in the sparse case or $y=\Phi(x)$35 in the non-sparse case, and assumption (A2), namely $y=\Phi(x)$36 (Feng, 2024).

5. Bayesian spike-and-slab sparse PCA

The Bayesian SP-SPCA of the 2021 paper is built on the spiked covariance model

$y=\Phi(x)$37

so marginally $y=\Phi(x)$38 with

$y=\Phi(x)$39

The columns of $y=\Phi(x)$40 are mutually orthogonal, and the paper adopts a jointly row-sparse loading matrix: each row is either identically zero across components or nonzero across all components. This support-sharing simplifies both posterior computation and theory (Ning et al., 2021).

The distinctive prior construction uses parameter expansion. An orthogonal latent matrix $y=\Phi(x)$41 is introduced and $y=\Phi(x)$42 is assigned a row-wise spike-and-slab prior, after which $y=\Phi(x)$43 is integrated out to induce the prior on $y=\Phi(x)$44. With slab family

$y=\Phi(x)$45

the paper allows product Laplace, multivariate Normal, and group-sparse slabs, together with Bernoulli-Beta inclusion priors and optional hyperpriors on $y=\Phi(x)$46 and $y=\Phi(x)$47. The technical purpose of the expansion is to circumvent the orthogonality constraint in posterior optimization while preserving the correct principal subspace target (Ning et al., 2021).

Inference is carried out by PX-CAVI, a parameter-expanded coordinate-ascent variational inference algorithm. The mean-field variational family assigns to each row a mixture of a point mass at zero and a Gaussian component,

$y=\Phi(x)$48

In the E-step, the latent scores have Gaussian variational posteriors with

$y=\Phi(x)$49

and

$y=\Phi(x)$50

The M-step is performed in an expanded parameter space $y=\Phi(x)$51, where $y=\Phi(x)$52 is a scaling expansion introduced to improve conditioning and accelerate convergence (Ning et al., 2021).

For the recommended multivariate Normal slab, the row-wise variational updates are closed form:

$y=\Phi(x)$53

The inclusion probability update is logistic via $y=\Phi(x)$54, and the noise variance update is also closed form under the Inverse-Gamma prior. PX “unwinding” then estimates $y=\Phi(x)$55, maps back to the unexpanded parameter, and recovers an orthogonal $y=\Phi(x)$56 by SVD so that the estimated subspace satisfies the required orthogonality (Ning et al., 2021).

The paper also develops a PX-EM analogue based on a continuous spike-and-slab prior. The acceleration claim is explicit: the parameter expansions reduce the spectral radius governing EM convergence, thereby speeding up iterations. For PX-CAVI, the per-iteration complexity under the Normal slab is

$y=\Phi(x)$57

with practical savings when many inclusion probabilities are near zero (Ning et al., 2021).

Theoretical results establish contraction rates for both the exact posterior and the variational posterior. Writing $y=\Phi(x)$58, the posterior contracts at this rate for the support size, covariance error $y=\Phi(x)$59, and subspace error $y=\Phi(x)$60, up to constants and the stated prior assumptions. The variational posterior contracts at the same rate. The paper characterizes these rates as near-optimal, differing from the classical sparse PCA minimax logarithmic factor only through $y=\Phi(x)$61 versus $y=\Phi(x)$62 (Ning et al., 2021).

Empirically, the multivariate Normal slab performs similarly to the Laplace slab at $y=\Phi(x)$63, but at $y=\Phi(x)$64 it yields smaller subspace error and substantially lower runtime. PX-CAVI generally outperforms PX-EM in subspace error and support recovery. Against elastic-net SPCA and robust SPCA, batch PX-CAVI achieves the smallest subspace error and the best support accuracy across the simulation grid. In the lung cancer gene-expression application with $y=\Phi(x)$65 subjects and $y=\Phi(x)$66 genes, $y=\Phi(x)$67 captured more than $y=\Phi(x)$68 variance; the top-$y=\Phi(x)$69 gene probe IDs for PC1 were identical across PX-CAVI, batch PX-CAVI, and PCA; PX-CAVI selected $y=\Phi(x)$70 active genes per PC, batch PX-CAVI used $y=\Phi(x)$71 genes for PC2, and PCA used all $y=\Phi(x)$72 genes (Ning et al., 2021).

6. SPCA-SP: fast deflation by subspace projections

The method called SPCA-SP is a sparse PCA algorithm based on subspace projections and Householder QR. In some communities it is referred to as SP-SPCA, but the paper itself uses SPCA-SP. Its central idea is not to solve a penalized sparse PCA criterion directly. Instead, it restricts each principal direction search to a low-dimensional subspace, extracts the leading dense direction there, applies post-hoc truncation to induce sparsity, and then updates the search subspace so that future dense directions are orthogonal to the previously truncated sparse loadings (Xu et al., 2019).

Let $y=\Phi(x)$73 be centered and $y=\Phi(x)$74. At round $y=\Phi(x)$75, with subspace projector $y=\Phi(x)$76 and $y=\Phi(x)$77, the dense direction is defined by

$y=\Phi(x)$78

The ambient-space proxy is $y=\Phi(x)$79, and sparsity is imposed by

$y=\Phi(x)$80

The method supports truncation-by-sparsity $y=\Phi(x)$81, truncation-by-energy $y=\Phi(x)$82, and hard-thresholding $y=\Phi(x)$83, each with explicit sparsity bounds in terms of its parameter (Xu et al., 2019).

The initial subspace $y=\Phi(x)$84 is produced by a randomized SVD procedure termed LinearTimeSVD. Sampling probabilities are

$y=\Phi(x)$85

rows are rescaled into a sampled matrix $y=\Phi(x)$86, and the left singular vectors of $y=\Phi(x)$87 are mapped back to form

$y=\Phi(x)$88

The one-off complexity is $y=\Phi(x)$89 (Xu et al., 2019).

The defining deflation step uses Householder QR. Form

$y=\Phi(x)$90

compute $y=\Phi(x)$91, and set

$y=\Phi(x)$92

Because $y=\Phi(x)$93 is upper triangular, the next projector satisfies

$y=\Phi(x)$94

Thus the next dense search space is exactly orthogonal to all previously computed sparse loadings. The fresh QR work at each round only involves the new block $y=\Phi(x)$95, with cost

$y=\Phi(x)$96

per round (Xu et al., 2019).

The paper also gives explicit post-truncation orthogonality bounds. If $y=\Phi(x)$97 and $y=\Phi(x)$98 are unit orthogonal vectors and $y=\Phi(x)$99, then

$k$00

Specializing to the three truncation rules yields bounds

$k$01

$k$02

and

$k$03

The paper uses these inequalities to formalize the tradeoff among truncation strength, sparsity, and near-orthogonality (Xu et al., 2019).

Runtime is one of the main motivations. Per component, the dominant costs are forming $k$04 in $k$05, computing $k$06 in $k$07, QR on a $k$08 block, and sorting for $k$09 or $k$10. Over $k$11 components the total complexity is

$k$12

plus the initial projection cost. The method does not materialize a full deflated $k$13 covariance across rounds (Xu et al., 2019).

The empirical section compares SPCA-SP with SPCA, PathSPCA, TPower, and SPCArt. On the Pitprops data with the balanced $k$14-$k$15-$k$16-$k$17-$k$18-$k$19 loading pattern, SPCA-SP with $k$20 and $k$21 achieved CPEV $k$22 and orthogonality $k$23, while SPCA had CPEV $k$24 and orthogonality $k$25, PathSPCA had $k$26 and $k$27, TPower had $k$28 and $k$29, and SPCArt had $k$30 and $k$31. On extra-high-dimensional random Gaussian data with $k$32 and $k$33 up to $k$34, SPCA-SP had much lower runtime than TPower and SPCArt while maintaining comparable CPEV and orthogonality (Xu et al., 2019).

7. Comparative interpretation and disambiguation

The five usages of SP-SPCA differ at the level of statistical model, optimization target, and output object. Sequential Principal Curves Analysis produces an explicit invertible nonlinear coordinate map and a density-aware metric; it is closest to nonlinear ICA and manifold unfolding, not to sparse eigenanalysis (Laparra et al., 2016). Single-Parametric Sparse Principal Component Analysis is a penalized matrix-factorization approach in which sparsity comes from $k$35 terms and stability comes from a direction-wise SPPCR penalty governed by a single scalar $k$36 (Hu et al., 14 Mar 2026). Spatial-sign SP-SPCA replaces covariance by a robust scatter surrogate defined from directions, with SSPCA solving a sparse leading-eigenvector problem under elliptical heavy-tailed models (Feng, 2024). Bayesian SP-SPCA treats sparse PCA as posterior inference in a spiked covariance model, using spike-and-slab priors, orthogonality-aware parameter expansion, and variational approximation (Ning et al., 2021). SPCA-SP is instead a greedy deflation-and-truncation algorithm whose orthogonality is maintained through subspace projections rather than through a penalized objective or probabilistic prior (Xu et al., 2019).

This multiplicity of meanings has two practical consequences. First, citations using the acronym alone are ambiguous unless the surrounding model class is specified. Second, the methods are not interchangeable: one should not infer robustness to heavy tails from the single-parametric method, nor posterior uncertainty quantification from SPCA-SP, nor sparse loading recovery from Sequential Principal Curves Analysis. A plausible implication is that the acronym has evolved locally within several subfields—nonlinear representation learning, robust high-dimensional PCA, penalized sparse PCA, Bayesian latent-factor modeling, and fast deflation algorithms—without a common naming convention.

Within that heterogeneous landscape, the shared theme is only broad: each method modifies classical PCA to address a limitation that ordinary eigenanalysis leaves unresolved. In the principal-curves version the limitation is linearity and global Euclidean geometry; in the single-parametric version it is unstable uniform penalization; in the spatial-sign version it is sensitivity to heavy tails and outliers; in the Bayesian version it is sparse subspace inference under orthogonality constraints; and in SPCA-SP it is computational cost in high dimensions (Laparra et al., 2016, Hu et al., 14 Mar 2026, Feng, 2024, Ning et al., 2021, Xu et al., 2019).