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Learning vector quantization

Learning vector quantization (LVQ) is a family of supervised prototype-based machine learning algorithms designed for statistical pattern classification, where codebook vectors (prototypes) are adjusted to represent class-conditional densities and define decision boundaries through Voronoi tessellation of the input space.^[1] Introduced by Teuvo Kohonen in the late 1980s as an extension of unsupervised vector quantization techniques, LVQ employs a competitive learning process to train prototypes by updating the "winner" prototype—the one closest to an input vector in Euclidean distance—based on whether the input is correctly classified, thereby approximating optimal Bayesian decision borders for classification tasks.^[2] The foundational variant, LVQ1, uses a simple Hebbian-style update rule where correctly classified winners are pulled toward the input and errors are pushed away, while subsequent algorithms like LVQ2 and LVQ3 introduce pairwise updates between winners and errors to refine boundaries more precisely. Over the decades, LVQ has evolved into numerous advanced variants, including generalized LVQ (GLVQ) for margin maximization, robust soft LVQ (RSLVQ) for probabilistic modeling, and kernel-based extensions like KGLVQ for non-linear separability, enhancing its applicability to complex datasets.^[1] These algorithms are noted for their interpretability, as prototypes directly represent learned class centers, and have demonstrated competitive performance against methods like support vector machines in multi-class problems, particularly in domains such as image processing, biomedical data analysis, and fault detection in industrial systems.^[1]

Introduction

Definition and overview

Learning vector quantization (LVQ) is a family of prototype-based supervised classification algorithms designed to learn a set of codebook vectors, or prototypes, that represent class regions in the input space for pattern recognition tasks. These prototypes serve as reference points to which input data vectors are assigned via nearest neighbor matching, enabling classification by determining the class label of the closest prototype. Unlike purely unsupervised methods, LVQ leverages labeled training data to refine the prototypes, ensuring they approximate optimal decision boundaries that minimize classification errors. At its core, LVQ treats prototypes as centroids for their respective classes, iteratively adjusting their positions to better capture the distribution of labeled samples. When a training vector is correctly classified by a prototype of the same class, the prototype is pulled slightly closer to the vector; conversely, if misclassified, the incorrect prototype is pushed away while the correct one may be attracted. This supervised adaptation enhances the separation of class-conditional densities, leading to more robust classifiers. LVQ builds on the principles of vector quantization (VQ), an unsupervised technique for data compression and clustering that maps input vectors to a finite set of representative code vectors to minimize reconstruction error. By incorporating class labels, LVQ transforms this unsupervised mapping into a discriminative process, improving boundary delineation between classes without requiring complex architectures like multilayer neural networks. For illustration, consider a two-dimensional feature space with two classes separated by a nonlinear boundary. Initial prototypes are placed roughly in class regions; during training, those near correctly classified points of the same class move toward them, while prototypes on the "wrong" side of the boundary are repelled from misclassified points, gradually sharpening the decision surface to reduce overlap.

Historical development

Learning vector quantization (LVQ) emerged in the late 1980s as a supervised learning extension of self-organizing maps (SOMs), developed by Finnish researcher Teuvo Kohonen to enable prototype-based classification in neural network architectures. Kohonen, a pioneer in associative memory and topographic mapping, introduced LVQ to address the limitations of unsupervised SOMs by incorporating class labels for discriminative tasks, marking a shift toward hybrid unsupervised-supervised paradigms in early neural computation. This innovation builds upon SOMs, which Kohonen had proposed in 1982 for feature mapping, but adapts them for targeted pattern recognition challenges prevalent in signal processing and speech analysis at the time. The foundational LVQ1 algorithm was first described in a 1986 technical report and detailed in Kohonen's 1988 publication in the journal Neural Networks, stemming from presentations at early neural computing conferences. This work laid the groundwork for competitive learning with labeled prototypes, gaining traction through subsequent discussions in neural computing forums. Refinements followed swiftly, with LVQ2 and LVQ3 introduced in Kohonen's 1990 paper at the International Joint Conference on Neural Networks (IJCNN), enhancing boundary adjustment and stability for more robust classification performance. These developments, spanning 1986–1990, positioned LVQ as a key contribution amid the explosive growth of connectionist models in the era's neural network renaissance. By the mid-1990s, LVQ had evolved from theoretical proposals to practical implementations, exemplified by the release of the LVQ-PAK software package in 1996, co-authored by Kohonen and collaborators at Helsinki University of Technology, which standardized algorithm application for research and engineering.^[3] Integration into mainstream machine learning toolboxes accelerated in the 2000s, with LVQ incorporated into MATLAB's Neural Network Toolbox around the early 2000s, facilitating widespread adoption in academic and industrial settings for tasks like phoneme recognition. During this period, LVQ played a pivotal role in advancing prototype-based classifiers, offering interpretable alternatives to emerging methods like support vector machines (introduced in 1995) and multilayer perceptrons, particularly in domains requiring few labeled examples and linear decision boundaries. Post-2010, LVQ has seen minor adaptations to interface with deep learning frameworks, such as recurrent variants for sequential data processing, yet its core principles of prototype learning and nearest-neighbor classification remain largely unchanged, underscoring its enduring simplicity and efficacy in an era dominated by end-to-end neural architectures. These updates, including generalized matrix LVQ extensions around 2015, reflect LVQ's niche influence in interpretable AI rather than widespread reinvention.^[4]

Mathematical foundations

Vector quantization principles

Vector quantization (VQ) is a classical technique in data compression and pattern recognition that maps high-dimensional input vectors from a continuous space to a finite set of discrete representative vectors, known as code vectors or prototypes, thereby approximating the probability distribution of the input data.^[5] This mapping enables efficient representation of large datasets by partitioning the input space into Voronoi regions, each associated with a prototype, which serves as the centroid of the data points assigned to it.^[5] The core process of VQ involves nearest-prototype matching, where each input vector is assigned to the closest prototype based on a distortion measure, most commonly the squared Euclidean distance.^[5] This assignment is followed by iterative minimization of the quantization error to refine the prototypes and achieve a better approximation of the data distribution.^[5] The quantization error, which quantifies the overall distortion introduced by the mapping, is formally defined as

E = \sum_{i=1}^N \| \mathbf{x}_i - \mathbf{w}_{c(i)} \|^2,

where \mathbf{x}_i denotes the i-th input vector, \mathbf{w}_{c(i)} is the prototype closest to \mathbf{x}_i, c(i) is the index of that prototype, and N is the total number of input vectors.^[5] Training of vector quantizers is typically performed using algorithms such as the generalized Lloyd algorithm, which underlies methods like k-means clustering and emphasizes competitive learning dynamics.^[6] In this framework, prototypes compete to represent subsets of the data: during each iteration, a winning prototype is selected for an input vector based on proximity, and it is then updated by moving closer to that vector, gradually forming a codebook that minimizes the average distortion.^[6] This iterative, unsupervised process converges to a local minimum of the quantization error, providing a foundational partitioning of the data space without requiring labeled information.^[6] Despite its effectiveness in data approximation and compression, pure VQ operates in an unsupervised manner and lacks inherent awareness of class structures in labeled data, which can lead to suboptimal performance in classification tasks where decision boundaries need to align with categorical distinctions. This limitation motivates the extension to supervised variants like learning vector quantization, which incorporate class labels to refine prototype placements.

Supervised classification framework

In the supervised classification framework of Learning Vector Quantization (LVQ), class labels from training data are integrated into the vector quantization process to enable discriminative learning. Each prototype, or codebook vector, is explicitly assigned to one of the predefined classes, transforming the unsupervised clustering of vector quantization into a labeled partitioning of the input space. An unseen input vector is then classified by assigning it to the class of its nearest prototype, determined via a distance metric such as the Euclidean norm, which effectively delineates class-conditional regions through Voronoi tessellation. This labeled assignment ensures that prototypes represent not just data clusters but decision-making boundaries tailored to classification tasks. The core of the supervised adjustment mechanism involves iteratively refining prototype positions using labeled inputs to enhance separation between classes. When an input vector is presented, the closest prototype (the winner) is updated conditionally on class agreement: if the input and prototype share the same class label, the prototype is pulled toward the input to better capture intra-class variations; conversely, if the labels differ, the prototype is pushed away from the input to widen inter-class margins and sharpen boundaries. These attract-and-repel dynamics, driven by a diminishing learning rate, progressively adapt the prototypes to minimize overlap between class regions while preserving compactness within classes. As a result, the framework forms nonlinear decision boundaries in the feature space, composed of piecewise linear hyperplanes midway between adjacent prototypes of opposing classes, allowing LVQ to model complex, non-convex class distributions without assuming linear separability. The general update rule for the winning prototype \mathbf{w}_j at time step t, given input \mathbf{x}(t) and learning rate \alpha(t), incorporates this conditional direction via a sign factor s:

\mathbf{w}_j(t+1) = \mathbf{w}_j(t) + s \cdot \alpha(t) \cdot \big( \mathbf{x}(t) - \mathbf{w}_j(t) \big)

where s = +1 for same-class attraction and s = -1 for different-class repulsion. This rule directly contrasts with unsupervised vector quantization's sole focus on error minimization, as LVQ prioritizes classification accuracy over distortion reduction. Evaluation in this framework centers on the misclassification rate, which measures the proportion of inputs incorrectly assigned to a prototype's class, serving as the primary objective unlike the mean squared quantization error in unsupervised settings. Prototypes are typically optimized until the error rate stabilizes, often assessed through hold-out validation or cross-validation to ensure generalization.

Core algorithms

LVQ1 algorithm

The LVQ1 algorithm, the original supervised variant of learning vector quantization introduced by Teuvo Kohonen, employs an iterative training procedure based on winner-take-all selection of the nearest prototype to refine class prototypes using labeled data. This simplest form focuses on adjusting a single winning prototype per input to attract it toward same-class samples and repel it from different-class samples, thereby sharpening decision boundaries without explicit optimization of a cost function.^[7] The algorithm begins with initialization of prototypes, typically one or more per class, placed in the input space using unsupervised methods such as k-means clustering to ensure reasonable starting positions that approximate the data distribution.^[7] Training proceeds sequentially through the labeled dataset over multiple epochs. For each input vector \mathbf{x}(t) with associated class label c(\mathbf{x}(t)), the winning prototype \mathbf{w}_j(t) is determined as the one minimizing the Euclidean distance \|\mathbf{x}(t) - \mathbf{w}_j(t)\|. If the winner's class c(\mathbf{w}_j(t)) matches c(\mathbf{x}(t)), the prototype is updated to move closer to the input via attraction:

\mathbf{w}_j(t+1) = \mathbf{w}_j(t) + \alpha(t) \left( \mathbf{x}(t) - \mathbf{w}_j(t) \right)

Conversely, if the classes mismatch, the prototype is repelled away from the input:

\mathbf{w}_j(t+1) = \mathbf{w}_j(t) - \alpha(t) \left( \mathbf{x}(t) - \mathbf{w}_j(t) \right)

All other prototypes remain unchanged in this step.^[7] Key parameters include the time-varying learning rate \alpha(t), which starts at an initial value (often around 0.1) and decreases monotonically—typically linearly or exponentially—to a small final value (e.g., 0.001) to promote stability as training progresses.^[7] Convergence is achieved empirically, often after 10–100 epochs, by monitoring the stability of classification error rates on held-out data, though the process can be sensitive to initialization and may require preprocessing for optimal results.^[7] This procedure directly incorporates supervision by leveraging class labels in the update decisions, extending the core vector quantization principles to classification tasks.^[7]

LVQ2 and LVQ3 variants

The LVQ2 algorithm (often specified as LVQ2.1) enhances the basic LVQ1 by incorporating paired updates involving the two closest prototypes to the input vector, enabling more precise boundary adjustments between classes. Updates are applied only when the two closest prototypes \mathbf{w}_j and \mathbf{w}_k (with j the winner) belong to different classes—one matching the input's class and the other not—and the input falls within a relative window around them. This condition ensures modifications target regions near decision boundaries. The window is defined by a parameter w (typically 0.2–0.3), such that the update occurs if \min\left(\frac{d_j}{d_k}, \frac{d_k}{d_j}\right) > \frac{1-w}{1+w}, where d_j and d_k are the distances from the input \mathbf{x} to the prototypes. In such cases, the prototype of the wrong class is repelled from \mathbf{x}, while the one of the correct class is attracted toward it:

\mathbf{w}_\text{wrong} \leftarrow \mathbf{w}_\text{wrong} - \alpha (\mathbf{x} - \mathbf{w}_\text{wrong}), \quad \mathbf{w}_\text{correct} \leftarrow \mathbf{w}_\text{correct} + \alpha (\mathbf{x} - \mathbf{w}_\text{correct})

where \alpha is the learning rate. This mechanism refines class boundaries more effectively than single-prototype updates in LVQ1, particularly in regions where prototypes of different classes are proximate.^[7] The LVQ3 variant builds on LVQ2 to improve stability and prevent excessive boundary overshooting during prolonged training. It applies the same paired updates for mixed-class prototype pairs as in LVQ2. Additionally, when both closest prototypes match the input's class, the winner is mildly attracted toward the input using a reduced learning rate \epsilon \alpha (with \epsilon typically 0.1–0.5) to consolidate same-class regions without major shifts:

\mathbf{w}_j \leftarrow \mathbf{w}_j + \epsilon \alpha (\mathbf{x} - \mathbf{w}_j)

The window parameter w is used similarly for the paired updates. Unlike LVQ2, which can be unstable for extended training, LVQ3 is designed for fine-tuning phases following LVQ1 or LVQ2 pretraining to maintain prototype stability without distorting established decision surfaces.^[7]

Extensions and variants

Optimized LVQ methods

Optimized Learning Vector Quantization (OLVQ) represents an advancement over earlier LVQ variants by replacing heuristic update rules with optimization of learning rates to better approximate gradient descent and minimize misclassification errors, enhancing convergence and classification accuracy. Introduced by Kohonen, OLVQ adjusts prototype vectors through updates similar to LVQ1, but with dynamically optimized learning rates for each codebook vector, determined locally to maximize the probability of correct classification for the current input. This approach mitigates issues in traditional LVQ, such as inconsistent updates, by tailoring the step sizes to balance attraction of correctly classified winners and repulsion of erroneous ones. Fuzzy Learning Vector Quantization (FLVQ) extends the LVQ framework by incorporating fuzzy set theory to handle class overlaps and ambiguous assignments, assigning partial memberships to multiple prototypes rather than hard winner-take-all decisions. In FLVQ, membership degrees are computed using a softmax-like function over distance-based similarities, allowing prototypes to share responsibility for a sample proportional to their proximity and class relevance; for input \mathbf{x}, the membership u_{jk} for prototype j and sample k is u_{jk} = \frac{\exp(-\|\mathbf{x}_k - \mathbf{w}_j\|^2 / \sigma)}{\sum \exp(-\|\mathbf{x}_k - \mathbf{w}_m\|^2 / \sigma)}, where \sigma controls fuzziness. Prototypes are then updated weighted by these memberships, promoting smoother boundaries and robustness to noisy data. This method, developed by Tsao et al., improves performance on datasets with inherent uncertainties by reducing overcommitment to single prototypes.^[8] Further optimizations include variants with adaptive learning rates and kernel integrations, such as Generalized Learning Vector Quantization (GLVQ), which optimizes a margin-based cost function to enhance generalization. In GLVQ, the objective is to minimize S = \sum_i f(\mu(\mathbf{x}_i)), where \mu(\mathbf{x}_i) = \frac{d_+ - d_-}{d_+ + d_-} measures the relative distance to the nearest correct prototype (d_+) versus incorrect one (d_-), and f is a sigmoid for robustness; updates use steepest descent: \Delta \mathbf{w}_+ = \eta \frac{f'(\mu)}{d_+ + d_-} (\mathbf{x} - \mathbf{w}_+), with analogous adjustment for \mathbf{w}_-. Proposed by Sato and Yamada, GLVQ supports non-linear mappings via kernels, making it suitable for complex decision boundaries. These optimized methods collectively reduce sensitivity to initial prototype placement and excel in high-dimensional spaces.^[9]^[1] Recent developments as of 2025 include recurrent variants of LVQ for handling sequential data, such as RecLVQ, which integrates recurrent structures into GLVQ for improved performance on time-series classification.^[4] Additionally, LVQ has been adapted for self-supervised learning settings to learn representations without labels, enhancing its utility in unsupervised tasks.^[10] Learning vector quantization (LVQ) shares foundational principles with other prototype-based techniques, which represent data classes through a set of learned prototypes in feature space and classify inputs by proximity to the nearest prototype. One closely related method is the self-organizing map (SOM), an unsupervised clustering algorithm that organizes high-dimensional data onto a low-dimensional grid while preserving topological relationships among inputs.^[11] Unlike SOM, which lacks class labels and focuses on exploratory data visualization and dimensionality reduction, LVQ incorporates supervision by assigning labels to prototypes and adjusting them based on labeled training examples to optimize decision boundaries for classification tasks.^[12] SOM can serve as a preprocessing step for LVQ, where its unsupervised prototypes are fine-tuned via supervised updates to enhance classification accuracy.^[12] Nearest prototype classifiers (NPCs) provide a broader framework encompassing LVQ, where classification relies on assigning an input to the class of its closest prototype under a defined distance metric, often Euclidean.^[13] LVQ variants represent adaptive training procedures within this NPC paradigm, learning prototypes iteratively to improve generalization, in contrast to static NPCs that use fixed centroids like class means.^[13] This distinguishes NPCs, including LVQ, from instance-based methods such as k-nearest neighbors (k-NN), which compare inputs to all training samples rather than a compact set of prototypes, leading to higher computational demands and less interpretability during inference.^[13] NPCs like LVQ emphasize margin maximization between prototypes, promoting robustness to perturbations compared to the density-based decisions in k-NN.^[13] Generalized LVQ (GLVQ) extends the standard LVQ framework by employing a margin-based cost function sensitive to class boundaries, updating prototypes via gradient descent on the relative distance between the nearest correct and incorrect prototypes.^[14] This approach addresses limitations in original LVQ algorithms, such as potential divergence, by ensuring convergence and better separability, particularly for overlapping classes, as demonstrated on character recognition tasks where GLVQ achieved near-perfect accuracy.^[14] GLVQ maintains the prototype-based core of LVQ but shifts focus from heuristic winner-take-all updates to a principled optimization that incorporates borderline sensitivity, making it a direct supervised enhancement over unsupervised clustering prototypes.^[14] A key distinction across these techniques lies in LVQ's emphasis on supervised adaptation of prototypes for discriminative tasks, contrasting with the unsupervised, topology-preserving nature of SOM or purely clustering-based prototypes.^[12] This supervised focus enables LVQ to outperform unsupervised methods in labeled classification while inheriting efficiency from prototype reduction.^[11] Post-2015, LVQ principles have influenced hybrid methods integrating prototypes into deep embeddings, such as deep LVQ (DLVQ), which combines neural feature extraction with GLVQ optimization to mitigate adversarial vulnerabilities in deep networks, achieving lower error rates on benchmarks like MNIST compared to softmax classifiers.^[15] These evolutions extend LVQ's prototype learning to scalable, end-to-end trainable systems in modern machine learning.^[15]

Applications and performance

Use in pattern recognition

Learning vector quantization (LVQ) serves as a primary tool in pattern recognition tasks, where it maps input feature vectors to class labels by associating them with the nearest prototype during classification. In handwritten digit recognition, LVQ quantizes feature vectors extracted from digit images, enabling effective mapping to digit classes on datasets akin to MNIST, which consist of grayscale images of handwritten numerals. This approach has been applied to classify digits by training prototypes on pixel-based or transformed features, demonstrating its utility in unsupervised-to-supervised transitions for digit identification.^[16] In speech recognition, LVQ facilitates phoneme clustering by learning labeled prototypes from acoustic feature vectors, such as mel-frequency cepstral coefficients, to distinguish speech units. Early applications achieved phoneme recognition accuracies of 80-90% in continuous speech contexts, with benchmarks in the 1990s reporting over 90% accuracy for isolated phonemes using LVQ variants integrated with hidden Markov models. These prototypes enable robust classification of phonetic segments, supporting larger vocabulary systems.^[17]^[18] For image analysis, LVQ quantizes pixel or feature vectors to perform texture classification and object recognition, grouping similar image regions based on statistical or transform-based descriptors like wavelet coefficients. In texture discrimination tasks, such as classifying material surfaces or corrosion patterns in industrial images, LVQ prototypes capture discriminative features, outperforming traditional clustering in labeled scenarios by adapting to class boundaries. This quantization reduces dimensionality while preserving textural variance for accurate categorization.^[19] In biomedical data analysis, LVQ has been applied to classify RNA sequences for distinguishing SARS-CoV-2 virus types, using prototypes to represent sequence features and achieve interpretable discrimination without alignment. It has also supported identification of parasitic larvae in veterinary diagnostics and handling imbalanced datasets in medical classification tasks.^[20]^[21]^[22] For fault detection in industrial systems, LVQ enables diagnosis of anomalies in equipment like proton exchange membrane fuel cells (PEMFCs) by learning prototypes from sensor data such as voltage and temperature, facilitating early detection of water management faults. Applications include real-time fault isolation in machines and condition monitoring of power transformers using multi-level LVQ for precise disk failure identification.^[23]^[24]^[25] Practical implementations of LVQ in pattern recognition often involve 10-100 prototypes per class to balance representational power and computational efficiency, depending on data complexity and class separability. Preprocessing steps, including normalization of feature vectors to unit length and dimensionality reduction via principal component analysis (PCA), enhance convergence and classification performance by mitigating scale variations and noise. Hybrids combining LVQ with PCA first project high-dimensional inputs to lower spaces before prototype learning, improving scalability for large datasets.^[26]^[27] A seminal case study is Teuvo Kohonen's development of the neural phonetic typewriter in the 1980s, which employed phonotopic maps refined by LVQ for real-time speech-to-text transcription. The system processed acoustic signals into phonemic sequences using LVQ-labeled prototypes, achieving 75-90% raw phoneme accuracy and 96-98% word recognition on a 1,000-word Finnish vocabulary after speaker adaptation with minimal samples. This hardware implementation on IBM PC/AT platforms demonstrated LVQ's feasibility for practical speech recognition systems in the pre-deep learning era.^[28]

Comparative advantages and limitations

Learning vector quantization (LVQ) offers several advantages over other classifiers, particularly in terms of interpretability and computational efficiency. The prototype-based nature of LVQ allows for straightforward visualization and inspection of decision boundaries, enabling domain experts to understand and validate classifications more easily than with black-box models like support vector machines (SVMs) or multilayer perceptrons.^[29] For medium-sized datasets, LVQ demonstrates faster training and prediction times compared to SVMs, as its Hebbian-style updates depend on a fixed number of prototypes rather than the full training sample size.^[30] Additionally, LVQ effectively captures nonlinear decision boundaries through adaptive prototype positioning, providing a balance of flexibility and simplicity without requiring kernel extensions.^[29] In comparisons with instance-based methods like k-nearest neighbors (k-NN), LVQ provides significant efficiency gains at test time by classifying inputs based on distance to a small set of prototypes (often one per class), rather than computing distances to all training examples, achieving speed-ups of 1–3 orders of magnitude while maintaining comparable accuracy.^[31] Relative to neural networks, LVQ is simpler to implement and interpret, avoiding the scalability challenges of deep architectures in terms of parameter tuning and resource demands, though it may underperform on very large or complex datasets.^[29] On standard benchmarks such as UCI datasets, LVQ variants typically achieve accuracies of 85–95%, with error rates 10–20% lower than unsupervised vector quantization alone due to supervised prototype refinement.^[30] Despite these strengths, LVQ has notable limitations. It is sensitive to initial prototype placement and outliers, which can lead to suboptimal convergence and requires careful initialization strategies for reliable performance.^[29] In very high-dimensional spaces, such as hyperspectral data, LVQ often proves suboptimal without incorporating kernel or relevance weighting extensions, showing reduced robustness compared to SVMs or radial basis function networks.^[32] Furthermore, without techniques like cross-validation, LVQ risks overfitting, especially in adaptive variants that adjust metrics dynamically.^[30] LVQ is best suited for labeled datasets of moderate size where explainability is prioritized over maximum accuracy on intricate nonlinear problems, such as in industrial sensor applications or pattern recognition tasks requiring human oversight.^[30]

References

[1]
A review of learning vector quantization classifiers - arXiv
Sep 23, 2015 · In this work we present a review of the state of the art of Learning Vector Quantization (LVQ) classifiers. A taxonomy is proposed which integrates the most ...<|control11|><|separator|>
[2]
Learning Vector Quantization | SpringerLink
Learning Vector Quantization. Download book PDF. Teuvo Kohonen. Part of the book series: Springer Series in Information Sciences ((SSINF,volume 30)). 2757 ...
[3]
(PDF) LVQ-PAK: The learning vector quantization program package
PDF | On Jan 1, 1996, Teuvo Kohonen and others published LVQ-PAK: The learning vector quantization program package | Find, read and cite all the research ...Missing: original | Show results with:original
[4]
Variants of recurrent learning vector quantization - ScienceDirect
Sep 1, 2022 · Generalized Learning Vector Quantization (GLVQ) is a prototype-based sparse classification model that can be trained using gradient descent ...Missing: definition | Show results with:definition
[5]
Vector Quantization and Signal Compression - SpringerLink
This book is devoted to the theory and practice of signal compression, ie , data compression applied to signals such as speech, audio, images, and video ...Missing: principles | Show results with:principles
[6]
[PDF] Least Squares Quantization in PCM
S. P. Lloyd and B. McMillan, “Linear least squares filtering and prediction of sampled signals,” in Proc. Symp. on Modern Network. The Hexagon Theorem. DONALD ...
[7]
[PDF] arXiv:1509.07093v1 [cs.LG] 23 Sep 2015
Sep 23, 2015 · In this paper we present a comprehensive review of the most relevant su- pervised LVQ algorithms developed since the original work of Teuvo ...
[8]
https://ieeexplore.ieee.org/document/714290
[9]
Generalized Learning Vector Quantization - NIPS papers
We propose a new learning method, "Generalized Learning Vec(cid:173) tor Quantization (GLVQ)," in which reference vectors are updated based on the steepest ...Missing: Optimized seminal
[10]
https://www.esann.org/sites/default/files/proceedings/2024/ES2024-47.pdf
[11]
[PDF] ECE656-Machine Learning and Adaptive Systems Lectures 25 & 26
Kohonen proposed three different types of LVQ algorithms. LVQ1 Algorithm: It is assumed that several codebook vectors, wis, have been established using.
[12]
[PDF] arXiv:2207.07208v1 [cs.LG] 14 Jul 2022
Jul 14, 2022 · Nearest prototype classifiers (NPCs) assign to each input point the label of the nearest proto- type with respect to a chosen distance metric. A.
[13]
None
### Definition and Equation for Cost Function in GLVQ
[14]
[PDF] Deep Learning Vector Quantization
We initialize the prototypes for DLVQ as the class conditional means in the embedding space. We start training with a large neighborhood, i.e. τ = 0.1, and ...Missing: influence post-
[15]
Recognizing images of handwritten digits using learning vector ...
This research is based on supervised learning vector quantization neural network categorized under artificial neural network. The images of digits are ...
[16]
[PDF] The self-organizing map - Proceedings of the IEEE
These strategies and learning algorithms were introduced by the present author [38], [43], [45] and called Learning Vector Quantization (LVQ). ... Teuvo Kohonen ( ...
[17]
A new HMM/LVQ hybrid algorithm for speech recognition
The average word accuracy for the original HMM-based system was 62%. When the LVQ classifier was incorporated, the word accuracy increased to 81%.< >.
[18]
A Texture Analysis Approach to Corrosion Image Classification
The classification is performed with a Learning Vector Quantization network and comparison is made with Gaussian and k-NN classifiers. The effectivity of the ...
[19]
Implementation of Principal Component Analysis and Learning ...
Aug 10, 2025 · the learning vector quantization network architecture. Parameter initialization is carried out for the training. model with a learning rate ( ...
[20]
Generalized relevance learning vector quantization - ScienceDirect
We propose a new scheme for enlarging generalized learning vector quantization (GLVQ) with weighting factors for the input dimensions.Missing: equation | Show results with:equation
[21]
[PDF] ISCA Archive
A flexible and inexpensive real-time speech recognitlon system is described. It operates in the following modes: recognitlon of isolated words from a large ...
[22]
Can Learning Vector Quantization be an Alternative to SVM...
Dec 17, 2016 · Learning vector quantization (LVQ) is one of the most powerful approaches for prototype based classification of vector data, ...
[23]
[PDF] Asymmetric Learning Vector Quantization for Efficient Nearest ...
Mar 24, 2017 · Empirical results exhibited superior performance of asymmetric generalized LVQ (GLVQ) over other state-of-the-art prototype generation methods ...<|control11|><|separator|>
[24]
Classification in high-dimensional spectral data: Accuracy vs ...
Aug 9, 2025 · We propose a new scheme for enlarging generalized learning vector quantization (GLVQ) with weighting factors for the input dimensions. The ...