The growing popularity of Contrastive Language-Image Pretraining (CLIP) has led to its widespread application in various visual downstream tasks. To enhance CLIP’s effectiveness and versatility, efficient few-shot adaptation techniques have been widely adopted. Among these approaches, training-free methods, particularly caching methods exemplified by Tip-Adapter, have gained attention for their lightweight adaptation without the need for additional fine-tuning. In this paper, we revisit Tip-Adapter from a kernel perspective, showing that caching methods function as local adapters and are connected to a well-established kernel literature. Drawing on this insight, we offer a theoretical understanding of how these methods operate and suggest multiple avenues for enhancing the Tip-Adapter baseline. Notably, our analysis shows the importance of incorporating global information in local adapters. Therefore, we subsequently propose a global method that learns a proximal regularizer in a reproducing kernel Hilbert space (RKHS) using CLIP as a base learner. Our method, which we call ProKeR (Proximal Kernel ridge Regression), has a closed form solution and achieves state-of-the-art performances across 11 datasets in the standard few-shot adaptation benchmark. Code is available at https://ybendou.github.io/ProKeR/

Model compression is a critical area of research in deep learning, particularly in vision, driven by the need to lighten models memory or computational footprints. While numerous methods for model compression have been proposed, most focus on pruning, quantization, or knowledge distillation. In this work, we delve into an under-explored avenue: reducing the resolution of the input image as a complementary approach to other types of compression. By systematically investigating the impact of input resolution reduction on both classification and semantic segmentation tasks and on convnets and transformer-based architectures, we demonstrate that this strategy provides an interesting alternative for model compression. Our experimental results on standard benchmarks highlight the potential of this method, achieving competitive performance while significantly reducing computational and memory requirements. This study establishes input resolution reduction as a viable and promising direction in the broader landscape of model compression techniques for vision applications.

In this study, we propose an implementation methodology of real-time few-shot learning on tiny FPGA SoCs such as the PYNQ-Z1 board with arbitrary fixed-point bit-widths. Tensil-based conventional design environments limited hardware implementations to fixed-point bit-widths of 16 or 32 bits. To address this, we adopt the FINN framework, enabling implementations with arbitrary bit-widths. Several customizations and minor adjustments are made, including: 1.Optimization of Transpose nodes to resolve data format mismatches, 2.Addition of handling for converting the final "reduce mean" operation to Global Average Pooling (GAP). These adjustments allow us to reduce the bit-width while maintaining the same accuracy as the conventional realization, and achieve approximately twice the throughput in evaluations using CIFAR-10 dataset.

Foundation models have transformed AI by reducing reliance on task-specific data through large-scale pretraining. While successful in language and vision, their adoption in EEG has lagged due to the heterogeneity of public datasets, which are collected under varying protocols, devices, and electrode configurations. Existing EEG foundation models struggle to generalize across these variations, often restricting pretraining to a single setup, resulting in suboptimal performance, in particular under linear probing. We present REVE (Representation for EEG with Versatile Embeddings), a pretrained model explicitly designed to generalize across diverse EEG signals. REVE introduces a novel 4D positional encoding scheme that enables it to process signals of arbitrary length and electrode arrangement. Using a masked autoencoding objective, we pretrain REVE on over 60,000 hours of EEG data from 92 datasets spanning 25,000 subjects, representing the largest EEG pretraining effort to date. REVE achieves state-of-the-art results on 10 downstream EEG tasks, including motor imagery classification, seizure detection, sleep staging, cognitive load estimation, and emotion recognition. With little to no fine-tuning, it demonstrates strong generalization, and nuanced spatio-temporal modeling. We release code, pretrained weights, and tutorials to support standardized EEG research and accelerate progress in clinical neuroscience.

BitPruning is a training method for minimizing inference bitlengths at any granularity while maintaining accuracy. BitPruning extends the meaning of fixed-point bitlenghts into the continuous domain by interpolating between the nearest two integers, enabling gradient descent to learn bitlengths together with other parameters. A novel regularizer penalizes large bitlength representations and can be modified to minimize other quantifiable criteria, such as number of operations or memory footprint. BitPruning learns thrifty representations while maintaining accuracy: With ImageNet, it produces an average per layer bitlength of 3.76 and 4.36 bits on ResNet18 and MobileNet V2 respectively, remaining within 0.5% of the base TOP-1 accuracy. Simple modifications of the BitPruning regularizer can be used to further reduce compute workload by up to 24%, as well as memory footprint in activation or weight-heavy tasks by up to 14% and 8% respectively.

In Machine Learning, Mixup is a data-dependent regularization technique that consists in creating virtual samples by linearly interpolating input signals and their associated outputs. It has been shown to significantly improve accuracy on standard datasets, in particular in the field of vision. However, authors have pointed out that Mixup can produce out-of-distribution virtual samples and even contradictions in the augmented training set, potentially resulting in adversarial effects. In this paper, we introduce Local Mixup in which distant input samples are weighted down when computing the loss. In constrained settings we demonstrate that Local Mixup can create a trade-off between bias and variance, with the extreme cases reducing to vanilla training and classical Mixup. Using standardized computer vision benchmarks, we also show that Local Mixup can improve test accuracy.

Transductive Few-Shot learning has gained increased attention nowadays considering the cost of data annotations along with the increased accuracy provided by unlabelled samples in the domain of few shot. Especially in Few-Shot Classification (FSC), recent works explore the feature distributions aiming at maximizing likelihoods or posteriors with respect to the unknown parameters. Following this vein, and considering the parallel between FSC and clustering, we seek for better taking into account the uncertainty in estimation due to lack of data, as well as better statistical properties of the clusters associated with each class. Therefore in this paper we propose a new clustering method based on Variational Bayesian inference, further improved by Adaptive Dimension Reduction based on Probabilistic Linear Discriminant Analysis. Our proposed method significantly improves accuracy in the realistic unbalanced transductive setting on various Few-Shot benchmarks when applied to features used in previous studies, with a gain of up to $6%$ in accuracy. In addition, when applied to balanced setting, we obtain very competitive results without making use of the class-balance artefact which is disputable for practical use cases.

We introduce the problem of Active Few-Shot Classification (AFSC) where the objective is to classify a small, initially unlabeled, dataset given a very restrained labeling budget. This problem can be seen as a rival paradigm to classical Transductive Few-Shot Classification (TFSC), as both these approaches are applicable in similar conditions. We first propose a methodology that combines statistical inference, and an original two-tier active learning strategy that fits well into this framework. We then adapt several standard vision benchmarks from the field of TFSC. Our experiments show the potential benefits of AFSC can be substantial, with gains in average weighted accuracy of up to 10% compared to state-of-the-art TFSC methods for the same labeling budget. We believe this new paradigm could lead to new developments and standards in data-scarce learning settings.

When dealing with signals, labeling a classification dataset implies to define classes that may approximate a smoother and more complicated ground truth. For example, natural images may contain multiple objects, only one of which is labeled in many vision datasets, or classes may result from the discretization of a regression problem where targets are continuous. Using cross-entropy to train deep models on such coarse labels is likely to roughly cut through the feature space, potentially disregarding the most meaningful such features, in particular losing information on the underlying fine-grain task. In this paper we are interested in the problem of solving fine-grain classification or regression, using a model trained on coarse-grain labels only. We show that standard cross-entropy can lead to overfitting to coarse-related features. We introduce an entropy-based regularization to promote more diversity in the feature space of trained models, and empirically demonstrate the efficacy of this methodology to reach better performance on the fine-grain problems. Our results are supported by theoretical developments and empirical validation.

BCI Motor Imagery datasets usually are small and have different electrodes setups. When training a Deep Neural Network, one may want to capitalize on all these datasets to increase the amount of data available and hence obtain good generalization results. To this end, we introduce a spatial graph signal interpolation technique, that allows to interpolate efficiently multiple electrodes. We conduct a set of experiments with five BCI Motor Imagery datasets comparing the proposed interpolation with spherical splines interpolation. We believe that this work provides novel ideas on how to leverage graphs to interpolate electrodes and on how to homogenize multiple datasets.

Finding good measures of robustness – i.e. the ability to correctly classify corrupted input signals – of a trained classifier is an important question for sensitive practical applications. In this paper, we point out that averaging the radius of robustness of samples in a validation set is a statistically weak measure. We propose instead to weight the importance of samples depending on their difficulty. We motivate the proposed score by a theoretical case study using logistic regression. We also empirically demonstrate the ability of the proposed score to measure robustness of classifiers with little dependence on the choice of samples in more complex settings, including deep convolutional neural networks and real datasets.

In machine learning, classifiers are typically susceptible to noise in the training data. In this work, we aim at reducing intra-class noise with the help of graph filtering to improve the classification performance. Considered graphs are obtained by connecting samples of the training set that belong to a same class depending on the similarity of their representation in a latent space. We show that the proposed graph filtering methodology has the effect of asymptotically reducing intra-class variance, while maintaining the mean. While our approach applies to all classification problems in general, it is particularly useful in few-shot settings, where intra-class noise can have a huge impact due to the small sample selection. Using standardized benchmarks in the field of vision, we empirically demonstrate the ability of the proposed method to slightly improve state-of-the-art results in both cases of few-shot and standard classification.

In most cases deep learning architectures are trained disregarding the amount of operations and energy consumption. However, some applications, like embedded systems, can be resource-constrained during inference. A popular approach to reduce the size of a deep learning architecture consists in distilling knowledge from a bigger network (teacher) to a smaller one (student). Directly training the student to mimic the teacher representation can be effective, but it requires that both share the same latent space dimensions. In this work, we focus instead on relative knowledge distillation (RKD), which considers the geometry of the respective latent spaces, allowing for dimension-agnostic transfer of knowledge. Specifically we introduce a graph-based RKD method, in which graphs are used to capture the geometry of latent spaces. Using classical computer vision benchmarks, we demonstrate the ability of the proposed method to efficiently distillate knowledge from the teacher to the student, leading to better accuracy for the same budget as compared to existing RKD alternatives.

In many application domains such as computer vision, Convolutional Layers (CLs) are key to the accuracy of deep learning methods. However, it is often required to assemble a large number of CLs, each containing thousands of parameters, in order to reach state-of-the-art accuracy, thus resulting in complex and demanding systems that are poorly fitted to resource-limited devices. Recently, methods have been proposed to replace the generic convolution operator by the combination of a shift operation and a simpler 1×1 convolution. The resulting block, called Shift Layer (SL), is an efficient alternative to CLs in the sense it allows to reach similar accuracies on various tasks with faster computations and fewer parameters. In this contribution, we introduce Shift Attention Layers (SALs), which extend SLs by using an attention mechanism that learns which shifts are the best at the same time the network function is trained. We demonstrate SALs are able to outperform vanilla SLs (and CLs) on various object recognition benchmarks while significantly reducing the number of float operations and parameters for the inference.

In few-shot classification, the aim is to learn models able to discriminate classes using only a small number of labeled examples. In this context, works have proposed to introduce Graph Neural Networks (GNNs) aiming at exploiting the information contained in other samples treated concurrently, what is commonly referred to as the transductive setting in the literature. These GNNs are trained all together with a backbone feature extractor. In this paper, we propose a new method that relies on graphs only to interpolate feature vectors instead, resulting in a transductive learning setting with no additional parameters to train. Our proposed method thus exploits two levels of information: a) transfer features obtained on generic datasets, b) transductive information obtained from other samples to be classified. Using standard few-shot vision classification datasets, we demonstrate its ability to bring significant gains compared to other works.

Because deep neural networks (DNNs) rely on a large number of parameters and computations, their implementation in energy-constrained systems is challenging. In this paper, we investigate the solution of reducing the supply voltage of the memories used in the system, which results in bit-cell faults. We explore the robustness of state-of-the-art DNN architectures towards such defects and propose a regularizer meant to mitigate their effects on accuracy. Our experiments clearly demonstrate the interest of operating the system in a faulty regime to save energy without reducing accuracy.

Transfer learning using deep neural networks as feature extractors has become increasingly popular over the past few years. It allows to obtain state-of-the-art accuracy on datasets too small to train a deep neural network on its own, and it provides cutting edge descriptors that, combined with nonparametric learning methods, allow rapid and flexible deployment of performing solutions in computationally restricted settings. In this paper, we are interested in showing that the features extracted using deep neural networks have specific properties which can be used to improve accuracy of downstream nonparametric learning methods. Namely, we demonstrate that for some distributions where information is embedded in a few coordinates, segmenting feature vectors can lead to better accuracy. We show how this model can be applied to real datasets by performing experiments using three mainstream deep neural network feature extractors and four databases, in vision and audio.

Quantifying the similarity between two networks is critical in many applications. A number of algorithms have been proposed to compute graph similarity, mainly based on the properties of nodes and edges. Interestingly, most of these algorithms ignore the physical location of the nodes, which is a key factor in the context of brain networks involving spatially defined functional areas. In this paper, we present a novel algorithm called "SimiNet" for measuring similarity between two graphs whose nodes are defined a priori within a 3D coordinate system. SimiNet provides a quantified index (ranging from 0 to 1) that accounts for node, edge and spatiality features. Complex graphs were simulated to evaluate the performance of SimiNet that is compared with eight state-of-art methods. Results show that SimiNet is able to detect weak spatial variations in compared graphs in addition to computing similarity using both nodes and edges. SimiNet was also applied to real brain networks obtained during a visual recognition task. The algorithm shows high performance to detect spatial variation of brain networks obtained during a naming task of two categories of visual stimuli: animals and tools. A perspective to this work is a better understanding of object categorization in the human brain.

We study an indexing architecture to store and search in a database of high-dimensional vectors from the perspective of statistical signal processing and decision theory. This architecture is composed of several memory units, each of which summarizes a fraction of the database by a single representative vector. The potential similarity of the query to one of the vectors stored in the memory unit is gauged by a simple correlation with the memory unit's representative vector. This representative optimizes the test of the following hypothesis: the query is independent from any vector in the memory unit versus the query is a simple perturbation of one of the stored vectors. Compared to exhaustive search, our approach finds the most similar database vectors significantly faster without a noticeable reduction in search quality. Interestingly, the reduction of complexity is provably better in high-dimensional spaces. We empirically demonstrate its practical interest in a large-scale image search scenario with off-the-shelf state-of-the-art descriptors.

Signal processing on graphs is a recent research domain that aims at generalizing classical tools in signal processing, in order to analyze signals evolving on complex domains. Such domains are represented by graphs, for which one can compute a particular matrix, called the normalized Laplacian. It was shown that the eigenvalues of this Laplacian correspond to the frequencies of the Fourier domain in classical signal processing. Therefore, the frequency domain is not the same for every support graph. A consequence of this is that there is no non-trivial generalization of Heisenberg's uncertainty principle, that states that a signal cannot be fully localized both in the time domain and in the frequency domain. A way to generalize this principle, introduced by Agaskar and Lu, consists in determining a curve that represents a lower bound on the compromise between precision in the graph domain and precision in the spectral domain. The aim of this paper is to propose a characterization of the signals achieving this curve, for a larger class of graphs than the one studied by Agaskar and Lu.

The problem of finding nearest neighbours in terms of Euclidean distance, Hamming distance or other distance metric is a very common operation in computer vision and pattern recognition. In order to accelerate the search for the nearest neighbour in large collection datasets, many methods rely on the coarse-fine approach. In this paper we propose to combine Product Quantization (PQ) and binary neural associative memories to perform the coarse search. Our motivation lies in the fact that neural network dimensions of the representation associated with a set of k vectors is independent of k. We run experiments on TEXMEX SIFT1M and MNIST databases and observe significant improvements in terms of complexity of the search compared to raw PQ.

Associative memories are data structures that allow retrieval of previously stored messages given part of their content. They, thus, behave similarly to the human brain's memory that is capable, for instance, of retrieving the end of a song, given its beginning. Among different families of associative memories, sparse ones are known to provide the best efficiency (ratio of the number of bits stored to that of the bits used). Recently, a new family of sparse associative memories achieving almost optimal efficiency has been proposed. Their structure, relying on binary connections and neurons, induces a direct mapping between input messages and stored patterns. Nevertheless, it is well known that nonuniformity of the stored messages can lead to a dramatic decrease in performance. In this paper, we show the impact of nonuniformity on the performance of this recent model, and we exploit the structure of the model to improve its performance in practical applications, where data are not necessarily uniform. In order to approach the performance of networks with uniformly distributed messages presented in theoretical studies, twin neurons are introduced. To assess the adapted model, twin neurons are used with the real-world data to optimize power consumption of electronic circuits in practical test cases.

An extension to a recently introduced architecture of clique-based neural networks is presented. This extension makes it possible to store sequences with high efficiency. To obtain this property, network connections are provided with orientation and with flexible redundancy carried by both spatial and temporal redundancies, a mechanism of anticipation being introduced in the model. In addition to the sequence storage with high efficiency, this new scheme also offers biological plausibility. In order to achieve accurate sequence retrieval, a double-layered structure combining heteroassociation and autoassociation is also proposed.

Associative memories are an alternative to classical indexed memories that are capable of retrieving a message previously stored when an incomplete version of this message is presented. Recently a new model of associative memory based on binary neurons and binary links has been proposed. This model named Clustered Neural Network (CNN) offers large storage diversity (number of messages stored) and fast message retrieval when implemented in hardware. The performance of this model drops when the stored message distribution is non-uniform. In this paper, we enhance the CNN model to support non-uniform message distribution by adding features of Restricted Boltzmann Machines. In addition, we present a fully parallel hardware design of the model. The proposed implementation multiplies the performance (diversity) of Clustered Neural Networks by a factor of 3 with an increase of complexity of 40%.

The focus of visual attention has been argued to play a key role in object recognition. Many computational models of visual attention were proposed to estimate locations of eye fixations driven by bottom-up stimuli. Most of these models rely on pyramids consisting of multiple scaled versions of the visual scene. This design aims at capturing the fact that neural cells in higher visual areas tend to have larger receptive fields (RFs). On the other hand, very few models represent multi-scaling resulting from the eccentricity-dependent RF sizes within each visual layer, also known as the cortical magnification effect. In this paper, we demonstrate that using a cortical-magnification-like mechanism can lead to performant alternatives to pyramidal approaches in the context of attentional modeling. Moreover, we argue that introducing such a mechanism equips the proposed model with additional properties related to overt attention and distance-dependent saliency that are worth exploring.

Associative memories are capable of retrieving previously stored patterns given parts of them. This feature makes them good candidates for pattern detection in images. Clustered Neural Networks is a recently-introduced family of associative memories that allows a fast pattern retrieval when implemented in hardware. In this paper, we propose a new pattern retrieval algorithm that results in a dramatically lower error rate compared to that of the conventional approach when used in oriented edge detection process. This function plays an important role in image processing. Furthermore, we present the corresponding hardware architecture and implementation of the new approach in comparison with a conventional architecture in literature, and show that the proposed architecture does not significantly affect hardware complexity.

We propose a low-power content-addressable memory (CAM) employing a new algorithm for associativity between the input tag and the corresponding address of the output data. The proposed architecture is based on a recently developed sparse clustered network using binary connections that on-average eliminates most of the parallel comparisons performed during a search. Therefore, the dynamic energy consumption of the proposed design is significantly lower compared with that of a conventional low-power CAM design. Given an input tag, the proposed architecture computes a few possibilities for the location of the matched tag and performs the comparisons on them to locate a single valid match. TSMC 65-nm CMOS technology was used for simulation purposes. Following a selection of design parameters, such as the number of CAM entries, the energy consumption and the search delay of the proposed design are 8%, and 26% of that of the conventional NAND architecture, respectively, with a 10% area overhead. A design methodology based on the silicon area and power budgets, and performance requirements is discussed.

Associative memories are data structures that allow retrieval of previously stored messages given part of their content. They thus behave similarly to human brain's memory that is capable for instance of retrieving the end of a song given its beginning. Among different families of associative memories, sparse ones are known to provide the best efficiency (ratio of the number of bits stored to that of bits used). Nevertheless, it is well known that non-uniformity of the stored messages can lead to dramatic decrease in performance. Recently, a new family of sparse associative memories achieving almost-optimal efficiency has been proposed. Their structure induces a direct mapping between input messages and stored patterns. In this work, we show the impact of non-uniformity on the performance of this recent model and we exploit the structure of the model to introduce several strategies to allow for efficient storage of non-uniform messages. We show that a technique based on Huffman coding is the most efficient.

We consider associative memories based on clustered graphs that were recently introduced. These memories are almost optimal in terms of the amount of storage they require (efficiency), and allow retrieving messages with low complexity. We study an unreliable implementation of the memory and compare its error rate and storage efficiency with that of a reliable implementation. We present analytical and simulation results that indicate that the proposed memory structure can tolerate a large number of faults at a reasonable cost, thereby making it a good candidate for achieving highly efficient circuit implementations of associative memories.

We study properties of the family of small-world random graphs introduced in Watts & Strogatz (1998), focusing on the spectrum of the normalized graph Laplacian. This spectrum influences the extent to which a signal supported on the vertices of the graph can be simultaneously localized on the graph and in the spectral domain (the surrogate of the frequency domain for signals supported on a graph). This characterization has implications for inferring or interpolating functions supported on such graphs when observations are only available at a subset of nodes.

A low-power Content-Addressable Memory (CAM) is introduced employing a new mechanism for associativity between the input tags and the corresponding address of the output data. The proposed architecture is based on a recently developed clustered-sparse network using binary-weighted connections that on-average will eliminate most of the parallel comparisons performed during a search. Therefore, the dynamic energy consumption of the proposed design is significantly lower compared to that of a conventional low-power CAM design. Given an input tag, the proposed architecture computes a few possibilities for the location of the matched tag and performs the comparisons on them to locate a single valid match. A 0.13μm CMOS technology was used for simulation purposes. The energy consumption and the search delay of the proposed design are 9.5%, and 30.4% of that of the conventional NAND architecture respectively with a 3.4% higher number of transistors.

Associative memories retrieve stored information given partial or erroneous input patterns. Recently, a new family of associative memories based on Clustered-Neural-Networks (CNNs) was introduced that can store many more messages than classical Hopfield-Neural Networks (HNNs). In this paper, we propose hardware architectures of such memories for partial or erroneous inputs. The proposed architectures eliminate winner-take-all modules and thus reduce the hardware complexity by consuming 65% fewer FPGA lookup tables and increase the operating frequency by approximately 1.9 times compared to that of previous work.

This paper considers the problem of inferring the structure of a network from indirect observations. Each observation (a “trace”) is the unordered set of nodes which are activated along a path through the network. Since a trace does not convey information about the order of nodes within the path, there are many feasible orders for each trace observed, and thus the problem of inferring the network from traces is, in general, ill-posed. We propose and analyze an algorithm which inserts edges by ordering each trace into a path according to which pairs of nodes in the path co-occur most frequently in the observations. When all traces involve exactly 3 nodes, we derive necessary and sufficient conditions for the reconstruction algorithm to exactly recover the graph. Finally, for a family of random graphs, we present expressions for reconstruction error probabilities (false discoveries and missed detections).

Associative memories are structures that store data in such a way that it can later be retrieved given only a part of its content - a sort-of error/erasure-resilience property. They are used in applications ranging from caches and memory management in CPUs to database engines. In this work we study associative memories built on the maximum likelihood principle. We derive minimum residual error rates when the data stored comes from a uniform binary source. Second, we determine the minimum amount of memory required to store the same data. Finally, we bound the computational complexity for message retrieval. We then compare these bounds with two existing associative memory architectures: the celebrated Hopfield neural networks and a neural network architecture introduced more recently by Gripon and Berrou.

Associative memories are alternatives to indexed memories that when implemented in hardware can benefit many applications such as data mining. The classical neural network based methodology is impractical to implement since in order to increase the size of the memory, the number of information bits stored per memory bit (efficiency) approaches zero. In addition, the length of a message to be stored and retrieved needs to be the same size as the number of nodes in the network causing the total number of messages the network is capable of storing (diversity) to be limited. Recently, a novel algorithm based on sparse clustered neural networks has been proposed that achieves nearly optimal efficiency and large diversity. In this paper, a proof-of-concept hardware implementation of these networks is presented. The limitations and possible future research areas are discussed.

We consider the problem of efficient and lossless representation of a multiset of m words drawn with repetition from a set of size 2n. One expects that encoding the (unordered) multiset should lead to significant savings in rate as compared to encoding an (ordered) sequence with the same words, since information about the order of words in the sequence corresponds to a permutation. We propose and analyze a practical multiset encoder/decoder based on the trie data structure. The act of encoding requires O(m(n + log m)) operations, and decoding requires O(mn) operations. Of particular interest is the case where cardinality of the multiset scales as m = 1/c2nfor some c >; 1, as n → ∞. Under this scaling, and when the words in the multiset are drawn independently and uniformly, we show that the proposed encoding leads to an arbitrary improvement in rate over encoding an ordered sequence with the same words. Moreover, the expected length of the proposed codes in this setting is asymptotically within a constant factor of 5/3 of the lower bound.

Coded recurrent neural networks with three levels of sparsity are introduced. The first level is related to the size of messages that are much smaller than the number of available neurons. The second one is provided by a particular coding rule, acting as a local constraint in the neural activity. The third one is a characteristic of the low final connection density of the network after the learning phase. Though the proposed network is very simple since it is based on binary neurons and binary connections, it is able to learn a large number of messages and recall them, even in presence of strong erasures. The performance of the network is assessed as a classifier and as an associative memory.