CantorNet: A Sandbox for Testing Geometrical and Topological Complexity Measures

Many natural phenomena are characterized by self-similarity, for example the symmetry of human faces, or a repetitive motif of a song. Studying of such symmetries will allow us to gain deeper insights into the underlying mechanisms of complex systems. Recognizing the importance of understanding these patterns, we propose a geometrically inspired framework to study such phenomena in artificial neural networks. To this end, we introduce \emph{CantorNet}, inspired by the triadic construction of the Cantor set, which was introduced by Georg Cantor in the $19^\text{th}$ century. In mathematics, the Cantor set is a set of points lying on a single line that is self-similar and has a counter intuitive property of being an uncountably infinite null set. Similarly, we introduce CantorNet as a sandbox for studying self-similarity by means of novel topological and geometrical complexity measures. CantorNet constitutes a family of ReLU neural networks that spans the whole spectrum of possible Kolmogorov complexities, including the two opposite descriptions (linear and exponential as measured by the description length). CantorNet's decision boundaries can be arbitrarily ragged, yet are analytically known. Besides serving as a testing ground for complexity measures, our work may serve to illustrate potential pitfalls in geometry-ignorant data augmentation techniques and adversarial attacks.

On the Sampling Sparsity of Neuromorphic Analog-to-Spike Conversion based on Leaky Integrate-and-Fire

In contrast to the traditional principle of periodic sensing neuromorphic engineering pursues a paradigm shift towards bio-inspired event-based sensing, where events are primarily triggered by a change in the perceived stimulus. We show in a rigorous mathematical way that information encoding by means of Threshold-Based Representation based on either Leaky Integrate-and-Fire (LIF) or Send-on-Delta (SOD) is linked to an analog-to-spike conversion that guarantees maximum sparsity while satisfying an approximation condition based on the Alexiewicz norm.

On the Solvability of the {XOR} Problem by Spiking Neural Networks

The linearly inseparable XOR problem and the related problem of representing binary logical gates is revisited from the point of view of temporal encoding and its solvability by spiking neural networks with minimal configurations of leaky integrate-and-fire (LIF) neurons. We use this problem as an example to study the effect of different hyper parameters such as information encoding, the number of hidden units in a fully connected reservoir, the choice of the leaky parameter and the reset mechanism in terms of reset-to-zero and reset-by-subtraction based on different refractory times. The distributions of the weight matrices give insight into the difficulty, respectively the probability, to find a solution. This leads to the observation that zero refractory time together with graded spikes and an adapted reset mechanism, reset-to-mod, makes it possible to realize sparse solutions of a minimal configuration with only two neurons in the hidden layer to resolve all binary logic gate constellations with XOR as a special case.

Geometrically Inspired Kernel Machines for Collaborative Learning Beyond Gradient Descent

This paper develops a novel mathematical framework for collaborative learning by means of geometrically inspired kernel machines which includes statements on the bounds of generalisation and approximation errors, and sample complexity. For classification problems, this approach allows us to learn bounded geometric structures around given data points and hence solve the global model learning problem in an efficient way by exploiting convexity properties of the related optimisation problem in a Reproducing Kernel Hilbert Space (RKHS). In this way, we can reduce classification problems to determining the closest bounded geometric structure from a given data point. Further advantages that come with our solution is that our approach does not require clients to perform multiple epochs of local optimisation using stochastic gradient descent, nor require rounds of communication between client/server for optimising the global model. We highlight that numerous experiments have shown that the proposed method is a competitive alternative to the state-of-the-art.

On Leaky-Integrate-and Fire as Spike-Train-Quantization Operator on Dirac-Superimposed Continuous-Time Signals

Leaky-integrate-and-fire (LIF) is studied as a non-linear operator that maps an integrable signal $f$ to a sequence $\eta_f$ of discrete events, the spikes. In the case without any Dirac pulses in the input, it makes no difference whether to set the neuron's potential to zero or to subtract the threshold $\vartheta$ immediately after a spike triggering event. However, in the case of superimpose Dirac pulses the situation is different which raises the question of a mathematical justification of each of the proposed reset variants. In the limit case of zero refractory time the standard reset scheme based on threshold subtraction results in a modulo-based reset scheme which allows to characterize LIF as a quantization operator based on a weighted Alexiewicz norm $\|.\|_{A, \alpha}$ with leaky parameter $\alpha$. We prove the quantization formula $\|\eta_f - f\|_{A, \alpha} < \vartheta$ under the general condition of local integrability, almost everywhere boundedness and locally finitely many superimposed weighted Dirac pulses which provides a much larger signal space and more flexible sparse signal representation than manageable by classical signal processing.

SNN Architecture for Differential Time Encoding Using Decoupled Processing Time

Spiking neural networks (SNNs) have gained attention in recent years due to their ability to handle sparse and event-based data better than regular artificial neural networks (ANNs). Since the structure of SNNs is less suited for typically used accelerators such as GPUs than conventional ANNs, there is a demand for custom hardware accelerators for processing SNNs. In the past, the main focus was on platforms that resemble the structure of multiprocessor systems. In this work, we propose a lightweight neuron layer architecture that allows network structures to be directly mapped onto digital hardware. Our approach is based on differential time coding of spike sequences and the decoupling of processing time and spike timing that allows the SNN to be processed on different hardware platforms. We present synthesis and performance results showing that this architecture can be implemented for networks of more than 1000 neurons with high clock speeds on a State-of-the-Art FPGA. We furthermore show results on the robustness of our approach to quantization. These results demonstrate that high-accuracy inference can be performed with bit widths as low as 4.

Quantization in Spiking Neural Networks

In spiking neural networks (SNN), at each node, an incoming sequence of weighted Dirac pulses is converted into an output sequence of weighted Dirac pulses by a leaky-integrate-and-fire (LIF) neuron model based on spike aggregation and thresholding. We show that this mapping can be understood as a quantization operator and state a corresponding formula for the quantization error by means of the Alexiewicz norm. This analysis has implications for rethinking re-initialization in the LIF model, leading to the proposal of 'reset-to-mod' as a modulo-based reset variant.

Spiking Neural Networks in the Alexiewicz Topology: A New Perspective on Analysis and Error Bounds

In order to ease the analysis of error propagation in neuromorphic computing and to get a better understanding of spiking neural networks (SNN), we address the problem of mathematical analysis of SNNs as endomorphisms that map spike trains to spike trains. A central question is the adequate structure for a space of spike trains and its implication for the design of error measurements of SNNs including time delay, threshold deviations, and the design of the reinitialization mode of the leaky-integrate-and-fire (LIF) neuron model. First we identify the underlying topology by analyzing the closure of all sub-threshold signals of a LIF model. For zero leakage this approach yields the Alexiewicz topology, which we adopt to LIF neurons with arbitrary positive leakage. As a result LIF can be understood as spike train quantization in the corresponding norm. This way we obtain various error bounds and inequalities such as a quasi isometry relation between incoming and outgoing spike trains. Another result is a Lipschitz-style global upper bound for the error propagation and a related resonance-type phenomenon.

Addressing Parameter Choice Issues in Unsupervised Domain Adaptation by Aggregation

We study the problem of choosing algorithm hyper-parameters in unsupervised domain adaptation, i.e., with labeled data in a source domain and unlabeled data in a target domain, drawn from a different input distribution. We follow the strategy to compute several models using different hyper-parameters, and, to subsequently compute a linear aggregation of the models. While several heuristics exist that follow this strategy, methods are still missing that rely on thorough theories for bounding the target error. In this turn, we propose a method that extends weighted least squares to vector-valued functions, e.g., deep neural networks. We show that the target error of the proposed algorithm is asymptotically not worse than twice the error of the unknown optimal aggregation. We also perform a large scale empirical comparative study on several datasets, including text, images, electroencephalogram, body sensor signals and signals from mobile phones. Our method outperforms deep embedded validation (DEV) and importance weighted validation (IWV) on all datasets, setting a new state-of-the-art performance for solving parameter choice issues in unsupervised domain adaptation with theoretical error guarantees. We further study several competitive heuristics, all outperforming IWV and DEV on at least five datasets. However, our method outperforms each heuristic on at least five of seven datasets.

Kernel Affine Hull Machines for Differentially Private Learning

This paper explores the use of affine hulls of points as a means of representing data via learning in Reproducing Kernel Hilbert Spaces (RKHS), with the goal of partitioning the data space into geometric bodies that conceal privacy-sensitive information about individual data points, while preserving the structure of the original learning problem. To this end, we introduce the Kernel Affine Hull Machine (KAHM), which provides an effective way of computing a distance measure from the resulting bounded geometric body. KAHM is a critical building block in wide and deep autoencoders, which enable data representation learning for classification applications. To ensure privacy-preserving learning, we propose a novel method for generating fabricated data, which involves smoothing differentially private data samples through a transformation process. The resulting fabricated data guarantees not only differential privacy but also ensures that the KAHM modeling error is not larger than that of the original training data samples. We also address the accuracy-loss issue that arises with differentially private classifiers by using fabricated data. This approach results in a significant reduction in the risk of membership inference attacks while incurring only a marginal loss of accuracy. As an application, a KAHM based differentially private federated learning scheme is introduced featuring that the evaluation of global classifier requires only locally computed distance measures. Overall, our findings demonstrate the potential of KAHM as effective tool for privacy-preserving learning and classification.