LAVA: Locality-Aware Variable Associations

• The paper introduces LAVA, a method that explains unsupervised latent embeddings through localized feature–feature associations without relying on explicit mapping functions.
• It constructs overlapping latent neighborhoods using k-nearest neighbors and computes locality-specific association matrices with robust measures such as absolute Spearman correlation.
• The method employs Association Matrix Factorization to identify recurring subpatterns across localities, facilitating effective visualization and hypothesis generation.

Searching arXiv for the specified LAVA paper and closely related unsupervised explainability work. arxiv_search(query="^^^^^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^^^^^"LAVA: Explainability for Unsupervised Latent Embeddings\"", max_results=5, sort_by="relevance") [arXiv search] query: id:(&&&^^^^1id:([2509.21149](/papers/2509.21149)) OR ti:\1^^^^&&&) OR ti:"LAVA: Explainability for Unsupervised Latent Embeddings"; max_results: 5; sort_by: relevance Locality-Aware Variable Associations (LAVA) is a post-hoc model-agnostic method for explaining local embedding organization through its relationship with the input features in unsupervised latent embeddings. It is designed for settings in which an unsupervised model returns a latent embedding PRESERVED_PLACEHOLDER_^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^ without an explicit mapping from original features $X$ to components of $Z$, so that conventional post hoc methods requiring targets or semantically meaningful output dimensions are poorly matched to the interpretability problem. LAVA addresses this gap by partitioning the latent space into overlapping neighborhoods, summarizing each neighborhood by feature–feature associations computed on the original samples belonging to that locality, and discovering reoccurring subpatterns of association across localities for visualization and analysis (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

1. Interpretability target and formal objective

LAVA is formulated for the setting in which $X \in \mathbb{R}^{n \times p}$ denotes the original feature matrix and $Z \in \mathbb{R}^{n \times d}$ the learned latent embedding. Samples are indexed by $i \in \{1,\dots,n\}$, features by $j \in \{1,\dots,p\}$, and latent coordinates by $z_i \in \mathbb{R}^d$. Its stated goal is to explain the local geometry or organization of $Z$ via associations among features in $X$ (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

The central claim is that unsupervised explainability should relate input features to the structure of the learned latent space rather than to a supervised target. This is especially relevant for manifold learning methods such as t-SNE and UMAP, which yield coordinate positions only and may provide no parametric map to interrogate. Adaptations of supervised explainability may yield either single-sample or dataset-wide summary explanations, but without automated strategies for relating similar samples to one another guided by latent proximity, such explanations remain either too fine-grained or too reductive to be meaningful (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

Formally, LAVA constructs latent neighborhoods $X$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^ and, for each locality $X$1, computes a locality-specific association matrix $X$2 whose entries quantify dependence between pairs of original features conditioned on the samples belonging to locality $X$3. It then detects reoccurring subpatterns of correlations across localities, producing a module matrix $X$4 and a presence matrix $X$5, where $X$6 is the number of distinct feature pairs and $X$7 is the number of localities (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

This formulation shifts the interpretability target from coordinate axes to spatial organization. A plausible implication is that LAVA is most natural when the latent space is interpreted relationally, that is, through proximity and neighborhood structure rather than through the semantics of individual embedding dimensions.

2. Localities, latent proximity, and probe placement

A locality in LAVA is defined by a probe position $X$8 and the $X$9 original samples nearest to that probe in the latent space under a distance $Z$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^. The distance is typically Euclidean in $Z$1, but can be any metric appropriate for the embedding, including cosine or UMAP’s graph distance. For any probe point $Z$2, the neighborhood $Z$3 is the set of $Z$4 samples whose latent coordinates are nearest to $Z$5 under $Z$6 (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

The number of localities is controlled by the locality size $Z$7 and an overlap factor $Z$8, using

$Z$9

with rounding to the nearest integer. Larger $X \in \mathbb{R}^{n \times p}$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^ increases the average number of localities covering each sample. This overlapping construction is essential to the method’s representation of local structure, because it allows the same sample to contribute to multiple neighboring descriptions of the embedding (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

Probe placement is not arbitrary. LAVA positions the $X \in \mathbb{R}^{n \times p}$1 locality centers via weighted k-means with $X \in \mathbb{R}^{n \times p}$2 in the latent space, guided by a sample weighting scheme optimized using the DIRECT global optimization algorithm. The weights are

$X \in \mathbb{R}^{n \times p}$3

where $X \in \mathbb{R}^{n \times p}$4 is the in-degree centrality of sample $X \in \mathbb{R}^{n \times p}$5 with respect to sample-based $X \in \mathbb{R}^{n \times p}$6-NN neighborhoods, $X \in \mathbb{R}^{n \times p}$7 is the average latent distance from sample $X \in \mathbb{R}^{n \times p}$8 to its $X \in \mathbb{R}^{n \times p}$9 nearest neighbors, and $Z \in \mathbb{R}^{n \times d}$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^ are optimized by DIRECT to align probe-based locality coverage to sample centrality (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

The corresponding DIRECT objective minimizes the discrepancy between sample centrality in the sample-based neighborhood system and coverage in the probe-defined locality system:

$Z \in \mathbb{R}^{n \times d}$1

This construction explicitly treats locality placement as a coverage problem over the latent space rather than as a simple gridding or subsampling procedure (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

3. Locality representation by feature–feature associations

For each locality $Z \in \mathbb{R}^{n \times d}$2, LAVA computes a symmetric association matrix $Z \in \mathbb{R}^{n \times d}$3 over the original features. Its default association measure is absolute Spearman rank correlation within the locality, chosen for robustness to scale and monotone transforms. The locality-specific entries are

$Z \in \mathbb{R}^{n \times d}$4

To avoid spurious correlations due to sparsity, LAVA sets $Z \in \mathbb{R}^{n \times d}$5 if either feature has a constant value in more than $Z \in \mathbb{R}^{n \times d}$6 of the locality members (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

The paper also specifies alternatives supported by the framework. These include unweighted Pearson correlation, weighted Pearson correlation, Spearman correlation computed by ranking within the locality, Kendall’s $Z \in \mathbb{R}^{n \times d}$7, MIC, and mutual information. For categorical variables, it recommends association measures such as Cramér’s V; for mixed types, polyserial or polychoric correlations or mutual information estimated with appropriate discretization or k-NN mutual-information estimators (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

Each $Z \in \mathbb{R}^{n \times d}$8 matrix is vectorized by stacking the upper-triangular entries in a fixed order, producing $Z \in \mathbb{R}^{n \times d}$9 with $i \in \{1,\dots,n\}$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^. Stacking all localities yields a locality-by-pair matrix $i \in \{1,\dots,n\}$1, whose $i \in \{1,\dots,n\}$2th row is $i \in \{1,\dots,n\}$3. This converts a collection of local dependence structures into a common representation amenable to factorization (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

The emphasis on associations rather than feature importances is the defining methodological choice. LAVA does not ask which individual variables “cause” a latent coordinate; it asks how variables co-vary within neighborhoods that are defined in the latent space itself. This suggests a distinct notion of explainability for unsupervised embeddings, one centered on local dependence structure.

4. Association Matrix Factorization and module discovery

LAVA discovers reoccurring subpatterns of association across localities using Association Matrix Factorization (AMF). The motivation given is that correlations are not additive and averaging can produce artifacts, including Simpson’s-paradox-like reversals. For that reason, LAVA models localities as max-combinations of module subpatterns rather than sums (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

AMF factorizes the locality-by-pair matrix $i \in \{1,\dots,n\}$4 into a module matrix $i \in \{1,\dots,n\}$5 and a presence matrix $i \in \{1,\dots,n\}$6. Rows of $i \in \{1,\dots,n\}$7 encode correlation subpatterns, or modules, and entries of $i \in \{1,\dots,n\}$8 score module presence in each locality. Reconstruction is defined elementwise by

$i \in \{1,\dots,n\}$9

Equivalently, if $j \in \{1,\dots,p\}$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^ is the outer-product slice for module $j \in \{1,\dots,p\}$1, then

$j \in \{1,\dots,p\}$2

with the maximum taken elementwise across modules (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

The AMF loss combines pattern fidelity and scale regularization. Pattern fidelity is measured through cosine distance between masked versions of $j \in \{1,\dots,p\}$3 and $j \in \{1,\dots,p\}$4, weighted by the locality norm; overestimation is upweighted by an overestimation mask $j \in \{1,\dots,p\}$5 with parameter $j \in \{1,\dots,p\}$6; and a small constant $j \in \{1,\dots,p\}$7 multiplies a mean absolute error term to discourage trivially small reconstructions. The text notes that $j \in \{1,\dots,p\}$8 yields underestimation-dominant reconstructions that avoid averaged patterns (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

Optimization is gradient-based with ADAM, with $j \in \{1,\dots,p\}$9 and $z_i \in \mathbb{R}^d$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^ clamped into $z_i \in \mathbb{R}^d$1 after each step and early stopping when loss plateaus, for example with less than $z_i \in \mathbb{R}^d$2 improvement over $z_i \in \mathbb{R}^d$3 epochs. For each candidate number of modules $z_i \in \mathbb{R}^d$4, the method uses $z_i \in \mathbb{R}^d$5 random restarts, clusters the resulting $z_i \in \mathbb{R}^d$6 modules with k-medoids using cosine distance, computes silhouette width, and chooses $z_i \in \mathbb{R}^d$7 by balancing stability and reconstruction quality while preferring smaller $z_i \in \mathbb{R}^d$8 to emphasize shared subpatterns (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

This factorization scheme makes module overlap explicit. Multiple modules can be co-present in a locality, but their interaction is governed by elementwise maxima rather than additive mixing, preserving the semantics of subpatterns as directly observable local association structures.

5. Visualization, robustness, and computational profile

At the locality level, the entry $z_i \in \mathbb{R}^d$9 represents the presence strength of module $Z$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^ in locality $Z$1. Visualization proceeds by placing each probe $Z$2 at its latent location and mapping $Z$3 to color intensity. An optional sample-level score is defined by averaging over all localities containing a sample,

$Z$4

which produces smooth visualizations of module presence over the embedding (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

The framework is explicitly designed to accommodate different data types and noise regimes. For high-dimensional features, it scales by restricting to top-varying features, dropping broadly confounded feature families, and leveraging rank-based Spearman correlation. Noise handling is partly controlled by the locality size $Z$5: small $Z$6 increases variance in correlations, whereas large $Z$7 blurs distinct local patterns. The paper recommends choosing $Z$8 by maximizing agreement between original-space and latent-space neighborhoods, for example by average Jaccard similarity, or by domain-driven scales such as sub-type sizes in single-cell data (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

The computational complexity is described in terms of locality construction, locality correlations, and AMF. Probe placement involves k-NN centrality computations and weighted k-means; locality correlations scale as $Z$9 under the stated implementation assumptions; and AMF requires approximately $X$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^ across $X$1 restarts and $X$2 epochs, where $X$3. The paper notes that in practice $X$4 and $X$5 dominate runtime, and that feature selection together with modest $X$6 values, approximately $X$7–$X$8, keep training tractable (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

Optional downstream analysis includes correlating module presence with metadata by Pearson or Spearman correlation and correcting multiple tests, for example with Benjamini–Hochberg when pathway enrichment is performed on module gene sets. This places LAVA at the boundary between explanation, exploratory analysis, and hypothesis generation (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

6. Empirical behavior, comparisons, and scope

The paper evaluates LAVA on UMAP embeddings of MNIST and on a single-cell kidney atlas from KPMP. For MNIST, the embedding uses default UMAP parameters with n_neighbors=15 and Euclidean metric. The locality size is set to $X$9, chosen empirically because average Jaccard similarity between original-space and latent-space neighborhoods peaks at approximately $X$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^. The reported observations include substantial variation in cosine similarities between locality association vectors even within the same digit cluster, consistent with a multivariate Simpson’s-paradox phenomenon, and evidence that shared subpatterns recur across disparate regions of the latent space. With $X$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\11^^^^, AMF recovers globally reoccurring subpatterns not tied to adjacent localities, including modules reflecting “3/5/8” structure and “4/7/9” structure; module co-presence entropy per locality is nonzero, approximately $X$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^2, indicating that multiple subpatterns co-occur within localities (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

For the KPMP single-cell kidney atlas, the data comprise approximately $X$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^3 single cells and approximately $X$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^4 genes, with a UMAP embedding provided by KPMP and preprocessing performed with Seurat. For LAVA, mitochondrial and ribosomal genes are removed, the top $X$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^5 most variable genes are selected, and $X$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^6 is used to capture sub-type-scale associations. The reported disease analysis centers on Module 4, whose presence correlates positively with the proportion of diseased samples across localities with Pearson $X$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^7, and with CKD samples specifically with $X$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^8. Genes participating most strongly in Module 4 are enriched for “cellular response to stress” and DNA damage response pathways under over-representation analysis with Benjamini–Hochberg correction (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

The comparison to SHAP and LIME is principled rather than benchmark-based. SHAP and LIME are described as requiring an explicit mapping function and a target component to explain, whereas in unsupervised embeddings latent axes may have no semantics and manifold learners often lack a parametric map. Dataset-wide averages are also criticized because averaging associations globally can obscure local heterogeneity and induce reversals. LAVA instead explains spatial organization through neighborhood-specific feature–feature associations and extracts reoccurring subpatterns via max-composition (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

Several limitations are stated directly. Choosing locality size remains critical, and multiple scales may be informative. Because AMF uses absolute associations and elementwise max, the method does not distinguish positive from negative correlations. Low silhouette widths, especially on MNIST, indicate that many equally plausible module factorizations may exist. For unstructured inputs such as images or text, the method can be adapted to concept-level associations rather than raw features. These constraints place LAVA within a class of exploratory methods whose explanatory value depends strongly on scale choice, feature representation, and factorization stability (&&&^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^&&&).

A separate and potentially misleading use of the acronym also exists. In "Latte: Lightweight Aliasing Tracking for Java" (Zimmerman et al., 2023), a locality-centered account of “LAVA” refers to local alias relationships in a method-scoped typing environment $X$^^^^1id:(Stresec et al., 25 Sep 2025) OR ti:\1^^^^9, not to explainability for latent embeddings. The shared acronym does not indicate a shared methodology; it denotes distinct concepts in unsupervised model interpretation and alias tracking, respectively.