Organizations: Graduate School of Informatics, Osaka Metropolitan University, Osaka 599-8531, Japan · Graduate School of Sustainable System Sciences, Osaka Metropolitan University, Osaka 599-8531, Japan
Self-Organizing Maps (SOM) are a classical method for unsupervised learning, vector quantization, and topographic mapping of high-dimensional data. However, existing SOM formulations often involve a trade-off between computational efficiency and a clearly defined optimization objective. Objective-based variants such as Soft Topographic Vector Quantization (STVQ) provide a principled formulation, but their neighborhood-coupled computations become expensive as the number of latent nodes increases. In this paper, we propose Self-Organizing Maps with Optimized Latent Positions (SOM-OLP), an objective-based topographic mapping method that introduces a continuous latent position for each data point. Starting from the neighborhood distortion of STVQ, we construct a separable surrogate local cost based on its local quadratic structure and formulate an entropy-regularized objective based on it. This yields a simple block coordinate descent scheme with closed-form updates for assignment probabilities, latent positions, and reference vectors, while guaranteeing monotonic non-increase of the objective and retaining linear per-iteration complexity in the numbers of data points and latent nodes. Experiments on a synthetic saddle manifold, scalability studies on the Digits and MNIST datasets, and 16 benchmark datasets show that SOM-OLP achieves competitive neighborhood preservation and quantization performance, favorable scalability for large numbers of latent nodes and large datasets, and the best average rank among the compared methods on the benchmark datasets.
Figures & tables
Method
Hyperparameter
Search range
Search scale
BSOM
σ
[10−3,103]
log
STVQf
σ
[10−3,103]
log
λ
[10−3,103]
log
GTM
m
{2,3,…,15}
linear
s
[10−3,103]
log
λreg
[10−3,103]
log
TABLE I: Hyperparameter search space for each method.
Method
TW ↑
CN ↑
QE ↓
Iters ↓
Time ↓ [ms/iteration]
BSOM
0.9945
0.9952
0.0089
17
0.40
STVQf
0.9945
0.9952
0.0057
34
4.09
GTM
0.9979
0.9986
0.0087
105
16.94
SOM-OLP
0.9986
0.9992
0.0102
110
3.86
TABLE II: Quantitative comparison on the saddle dataset. Bold : best; underlined : second best.
Fig. 1: Comparison of BSOM, STVQf, GTM, and SOM-OLP on the saddle dataset. For each method, the data-space view shows the learned reference vectors, while the latent-space view shows the corresponding representation of the data points. The same color coding based on the original x3 values is used in both spaces to facilitate visual comparison of neighborhood relationships across the two spaces.
Method
TW ↑
CN ↑
QE ↓
Iters ↓
Time ↓ [ms/iteration]
BSOM
0.9871
0.9657
1.2217
21
11.45
STVQf
0.9872
0.9685
0.9315
33
25.96
GTM
0.9910
0.9746
1.1518
89
58.85
SOM-OLP
0.9864
0.9761
0.8895
37
29.77
TABLE III: Quantitative comparison on the Digits dataset. Bold : best; underlined : second best.
Fig. 2: Latent representations of the Digits dataset. While BSOM and STVQf are constrained to discrete nodes, GTM and SOM-OLP provide continuous latent positions. Notably, SOM-OLP exhibits a more adaptive distribution that reflects the local density of the data manifold.
M
BSOM
STVQf
GTM
SOM-OLP
2500=502
81
263
288
265
10000=1002
359
1 173
1 143
1 064
22500=1502
975
3 247
3 324
2 430
40000=2002
4 018
10 356
8 268
4 641
62500=2502
22 030
157 937
55 339
8 617
90000=3002
OOM
OOM
OOM
12 988
TABLE IV: Per-iteration wall-clock time (ms) versus grid size on the Digits dataset ( T=20 iterations). OOM: out-of-memory error.
Fig. 3: Latent representation of the MNIST dataset obtained by SOM-OLP with a 16×16 grid. Data points are colored by their ground-truth digit labels.
Dataset
N
D
PCA
BSOM
STVQf
GTM
SOM-OLP
Iris
150
4
0.9820
0.9799
0.9838
0.9787
0.9869
Wine
178
13
0.9041
0.9503
0.9531
0.9595
0.9593
Sonar
208
60
0.8754
0.9277
0.9391
0.9395
0.9344
Seeds
210
7
0.9615
0.9730
0.9751
0.9700
0.9806
Glass
214
9
0.8884
0.9527
0.9490
0.9574
0.9497
Heart Disease
303
13
0.8348
0.9222
0.9265
0.9317
0.9350
TABLE V: (TW+CN)/2 on benchmark datasets ( M=16×16 , standardized). Bold : best; underlined : second best.
Fig. 4: Critical-difference diagram based on the average ranks of (TW+CN)/2 over the 16 benchmark datasets.
GPU-accelerated Self-Organizing Map (SOM) implementations are among the most competitive options for large-scale SOM analysis, but growing dataset sizes increasingly challenge their practical use because workloads no longer fit cleanly within device-memory limits. We introduce FloatSOM, a SOM framework for scalable training and deployment that supports multi-GPU execution, out-of-memory disk-backed streaming, and novel topologies beyond regular lattices. We evaluate FloatSOM on 14 synthetic and real benchmark datasets together with controlled speed scaling benchmarks, and show that these improved topologies, combined with topology-aware hyperparameter fine-tuning, yield lower quantization error than current state-of-the-art SOM baselines. FloatSOM also sustains this performance at large scale with high-throughput distributed execution; in the largest benchmark, it trains a 1024-node SOM network on 1,000,000,000 samples with 50 features in 6.16 minutes on 8 GPUs across two separate high-performance-computing nodes.
Tony Xu, Sarah Klamt, Katherine Turner +3
John Curtin School of Medical Research, Australian National University · Department of Information Technology and Electrical Engineering, ETH Zürich · Mathematical Data Science Centre, Australian National University +2
Self-Organizing Maps (SOMs) have long been used as exploratory tools for high-dimensional data: they organize objects into a two-dimensional topology that reveals clusters, gradients, sparse regions, dense regions, and boundaries. Yet, in modern data systems, SOMs are typically trained and visualized outside the DBMS, disconnected from the relational data they summarize. We introduce the abstraction of a queryable data map: a learned topological artifact consisting of representatives, neighborhood relations, object assignments, and derived summaries. We instantiate this idea with MapDB, a lightweight prototype that makes SOM artifacts queryable so users can explore data topology without leaving the database. Experimental study shows that SOM training is feasible at moderate analytical scale, that map queries are interactive after materialization, and that SOM regions provide meaningful targets for exploratory SQL.
Denis Mayr Lima Martins, Gottfried Vossen
Department of Computing and Mathematics University of Sao Paulo Ribeirão Preto, São Paulo, Brazil · Department of Information Systems University of Münster Münster, Germany
Dense self-expression matrices and full-affinity spectral clustering limit the scalability of subspace clustering. We introduce the Latent Orthogonal Optimization Model for Subspace Clustering (LoomSC), a framework that addresses both bottlenecks through projector factorization and exact spectral reduction. Motivated by the spectral structure of least-squares regression, LoomSC jointly learns latent features and a projector self-representation through two thin factors. Alternating Procrustes and least-squares updates preserve the sample factor's orthogonality while keeping the coefficient matrix implicit. We construct a nonnegative quadratic affinity that preserves the projector's support. An exact feature map then reduces its normalized spectral problem to an eigenproblem whose dimension depends only on the factor width. Neither the full affinity nor the sample Laplacian needs to be formed. Our analysis quantifies the projector approximation and identifies conditions for subspace preservation and within-subspace connectivity. For fixed dimensions and iteration budgets, the complete pipeline has linear time and memory complexity in the number of samples. Across five image-clustering benchmarks, LoomSC ranks first or second in all 15 dataset-metric comparisons against 9 state-of-the-art baselines. Its mean accuracy exceeds the highest baseline mean by 6.66 percentage points. Synthetic experiments scale to 500,000 samples while maintaining at least 99.8% accuracy.
École de technologie supérieure (ÉTS), Montréal, QC, Canada · University of Bahrain (UOB), Kingdom of Bahrain · Université du Québec à Montréal (UQAM), Montréal, QC, Canada +1