Algebraic Latent Projection

• Algebraic latent projection is a set of techniques that use closed-form algebraic operators to decipher and manipulate latent representations in machine learning and signal processing.
• It extends classical PCA by using polynomial equations and debiased moment matrices to handle noise, nonlinear settings, and cross-model transfer tasks.
• The method leverages eigendecomposition of constructed moment matrices to extract latent features, ensuring robust statistical guarantees such as O_P(n⁻¹ᐟ²) convergence and effective SVD updates.

Algebraic latent projection encompasses a family of techniques that use algebraic, often closed-form, projection operators to decipher, manipulate, align, or update latent representations in machine learning and signal processing. Unlike purely geometric or iterative approaches, algebraic latent projection methods exploit the underlying structure of the data or model—often expressed via polynomial equations, subspace projectors, or anchor-based mappings—to solve inference, disentanglement, translation, or update tasks in a latent (hidden) space. These procedures extend the reach of standard matrix factorization and projection (such as principal component analysis) to nonlinear settings, cross-model transfer, online matrix updating, and disentangled representation learning.

1. Algebraic Structure of Latent Spaces

A latent space is often endowed with additional algebraic structure beyond mere vector-space linearity. In the classical PCA setting, the latent variables $\theta_1, ..., \theta_n \in \mathbb{R}^d$ are constrained to an affine subspace $L$, defined as the zero set of $k$ linear forms: $$L = \{x \in \mathbb{R}^d : b_1^\top x = \cdots = b_k^\top x = 0\}.$$ Algebraic latent projection generalizes this by considering algebraic sets $\mathcal{A} \subset \mathbb{R}^d$ specified as the common zero locus of real polynomial equations: $$\mathcal{A} = \mathcal{V}(P_1, ..., P_K) = \{x \in \mathbb{R}^d : P_j(x) = 0, \;\forall j=1,...,K\},$$ where each $P_j \in \mathbb{R}[x_1, ..., x_d]$. The problem is then to recover $\mathcal{A}$, or to identify and manipulate latent factors satisfying such nonlinear constraints, from observed or perturbed data (González-Sanz et al., 4 Aug 2025).

2. Moment Matrix Construction and Debiasing

Given noisy observations $X_i = \theta_i + \epsilon_i$, with $\epsilon_i$ i.i.d. Gaussian, algebraic latent projection proceeds by embedding the data points in a high-dimensional polynomial (Veronese) space using the degree-$g$ Veronese map,

$$\phi_{d,g}(x) = \{x_1^{i_1}\cdots x_d^{i_d}\}_{i_1+\dots+i_d \leq g} \in \mathbb{R}^{\kappa_{d,g}},\;\;\kappa_{d,g} = \binom{d+g}{d}.$$

The empirical Vandermonde matrix $V_n$ collects these mapped points, and the (biased) empirical moment matrix is $M_n = n^{-1} V_n^\top V_n$. In the absence of noise, the kernel of $M_n$ reveals the coefficients of all polynomials of degree $\leq g$ vanishing on the data.

To rigorously account for bias induced by noise, a debiased moment matrix $\widetilde{M}_n$ is constructed via an explicit tensor-moment expansion. The unbiased estimator uses alternating sign corrections and known noise covariance $\Sigma$: $$\widetilde{M}_n = \frac{1}{n}\sum_{i=1}^n \sum_{k=0}^g C_{2g, k} (-1)^k h \circ (\gamma \otimes \gamma) \text{sym}(\tilde{X}_i^{\otimes (2g - 2k)} \otimes \tilde{\Sigma}^{\otimes k}),$$ ensuring that $\mathbb{E}[\widetilde{M}_n]$ equals the noiseless population moment matrix (González-Sanz et al., 4 Aug 2025).

3. Extraction of Algebraic Features via Spectral Kernel

Algebraic latent projection harnesses eigendecomposition of $\widetilde{M}_n$ to identify the algebraic structure. The eigenvectors associated with near-zero eigenvalues span an estimate $\widehat{J}$ of the kernel, supplying consistent estimators for the coefficients of vanishing polynomials. Each $u_{j,n}$ is readily interpreted as the coefficient vector of a polynomial: $$\widehat{P}_j(x) = \sum_{|\alpha|\leq g} (U_0)_{\alpha, j}\;x^\alpha.$$ Under regularity assumptions (finite moments, completeness of intersection), subspace convergence and $\sqrt{n}$-consistency are guaranteed; that is, the Hausdorff error between true and estimated algebraic set decays as $O_P(n^{-1/2})$ locally, with asymptotically normal coefficient estimation (González-Sanz et al., 4 Aug 2025).

4. Algorithmic Strategies and Statistical Guarantees

The general workflow for algebraic latent projection in the context of algebraic set estimation follows three principal steps:

1. Embed sample points via a Veronese map and construct $\widetilde{M}_n$.
2. Perform eigendecomposition and select kernel vectors corresponding to vanishing eigenvalues.
3. Interpret kernel vectors as polynomial generators and reconstruct the target set.

Three reconstruction schemes are available:

• Zero-locus estimator: Directly solve for the common zero set of the learned polynomials. Local and global convergence results hold under regularity.
• Semi-algebraic tube estimator: For a threshold $\lambda_n$, define a tube $$\mathcal{A}_{n,\mathrm{tube}} = \{x : |\widehat{P}_j(x)| \leq \lambda_n\;\forall j\}$$. This method achieves convergence without structure assumptions, with tube radius scaling as $O_P((\log n)/\sqrt{n})$ for single-tuning procedures.
• Structure-aware projection: When prior structure is available, project kernel vectors onto constraint sets defined by domain knowledge (e.g., polynomials factoring as products of lines). Consistency is preserved under minimal regularity (González-Sanz et al., 4 Aug 2025).

5. Applications Beyond Polynomial Varieties

Algebraic latent projection also underpins methodologies in other contexts:

• Subspace factorization and disentanglement: In autoencoding setups, matrix subspace projection uses projectors onto labeled attribute subspaces to disentangle attribute and residual representations. The technique defines

$$P_{\mathrm{attr}} = A(A^\top A)^{-1}A^\top,\;\;P_{\mathrm{res}} = I - P_{\mathrm{attr}}$$

to algebraically separate and swap attribute components, enabling controlled manipulation and transfer of factors in latent space (Li et al., 2019).

• Model translation and stitching: Inverse relative projection maps representations between independently trained models via isometric, angle-preserving projections through a shared anchor-defined space. Algebraic invertibility properties (full-rank anchors, scale invariance in decoders) guarantee accurate, closed-form translation between latent spaces, facilitating zero-shot cross-model stitching and cross-modal transfer (Maiorca et al., 2024).
• Low-rank matrix SVD updating: The algebraic-projection view is critical in truncated SVD maintenance for evolving matrices. Projective subspaces, possibly augmented with resolvent corrections, enable efficient, high-accuracy updates of latent semantic spaces in streaming or dynamic settings by leveraging prior SVD factors and projecting onto augmented block subspaces or resolving spectral corrections (Kalantzis et al., 2020).

6. Comparative Overview of Methods and Performance

The following table summarizes key differences across algebraic latent projection frameworks as documented in the literature:

Paper / Setting	Projection Type	Core Guarantee
(González-Sanz et al., 4 Aug 2025)	Veronese/Vandermonde + debias	$O_P(n^{-1/2})$ estimation; Hausdorff and PK convergence
(Li et al., 2019)	Orthogonal subspace via $A$	Exact attribute disentanglement, no adversaries
(Maiorca et al., 2024)	Anchor-based, angle preserving	High cosine similarity (0.85–0.97); invertible mapping
(Kalantzis et al., 2020)	Rayleigh–Ritz projection/SVD	Near-optimal Ritz error for updated SVD

Performance validation includes, for example, $0.85$–$0.97$ cosine similarity in cross-latent translation and sub-$10^{-6}$ relative error for retained SVD singular modes, as reported in empirical studies (Maiorca et al., 2024, Kalantzis et al., 2020).

7. Extensions, Generalizations, and Outlook

While foundational approaches rely on linear or polynomial projections, generalizations are proposed to address cases where decoders are only approximately isometric, or where nonlinear invertible mappings (kernel anchors, neural refinements) are required. Adaptive anchor schemes and dynamic ensembles offer improvements in numerical stability and conditioning for model stitching frameworks. For latent space factorization, the purely algebraic approach shows robustness across modalities (images, text), and scalability in both dynamic and high-dimensional regimes (González-Sanz et al., 4 Aug 2025, Maiorca et al., 2024, Kalantzis et al., 2020, Li et al., 2019).

A plausible implication is that algebraic latent projection provides a principled toolkit unifying several seemingly disparate tasks—algebraic set recovery, SVD updating, latent disentanglement, and cross-model translation—under a shared projection-based framework, with rigorous statistical and computational guarantees.