Controlling the parameters' norm often yields good generalisation when training neural networks. Beyond simple intuitions, the relation between regularising parameters' norm and obtained estimators remains theoretically misunderstood. For one hidden ReLU layer networks with unidimensional data, this work shows the parameters' norm required to represent a function is given by the total variation of its second derivative, weighted by a $\sqrt{1+x^2}$ factor. Notably, this weighting factor disappears when the norm of bias terms is not regularised. The presence of this additional weighting factor is of utmost significance as it is shown to enforce the uniqueness and sparsity (in the number of kinks) of the minimal norm interpolator. Conversely, omitting the bias' norm allows for non-sparse solutions. Penalising the bias terms in the regularisation, either explicitly or implicitly, thus leads to sparse estimators.
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For the convolutional neural network (CNN) used for pattern classification, the training loss function is usually applied to the final output of the network, except for some regularization constraints on the network parameters. However, with the increasing of the number of network layers, the influence of the loss function on the network front layers gradually decreases, and the network parameters tend to fall into local optimization. At the same time, it is found that the trained network has significant information redundancy at all stages of features, which reduces the effectiveness of feature mapping at all stages and is not conducive to the change of the subsequent parameters of the network in the direction of optimality. Therefore, it is possible to obtain a more optimized solution of the network and further improve the classification accuracy of the network by designing a loss function for restraining the front stage features and eliminating the information redundancy of the front stage features .For CNN, this article proposes a multi-stage feature decorrelation loss (MFD Loss), which refines effective features and eliminates information redundancy by constraining the correlation of features at all stages. Considering that there are many layers in CNN, through experimental comparison and analysis, MFD Loss acts on multiple front layers of CNN, constrains the output features of each layer and each channel, and performs supervision training jointly with classification loss function during network training. Compared with the single Softmax Loss supervised learning, the experiments on several commonly used datasets on several typical CNNs prove that the classification performance of Softmax Loss+MFD Loss is significantly better. Meanwhile, the comparison experiments before and after the combination of MFD Loss and some other typical loss functions verify its good universality.
Deep feedforward and recurrent rate-based neural networks have become successful functional models of the brain, but they neglect obvious biological details such as spikes and Dale's law. Here we argue that these details are crucial in order to understand how real neural circuits operate. Towards this aim, we put forth a new framework for spike-based computation in low-rank excitatory-inhibitory spiking networks. By considering populations with rank-1 connectivity, we cast each neuron's spiking threshold as a boundary in a low-dimensional input-output space. We then show how the combined thresholds of a population of inhibitory neurons form a stable boundary in this space, and those of a population of excitatory neurons form an unstable boundary. Combining the two boundaries results in a rank-2 excitatory-inhibitory (EI) network with inhibition-stabilized dynamics at the intersection of the two boundaries. The computation of the resulting networks can be understood as the difference of two convex functions and is thereby capable of approximating arbitrary non-linear input-output mappings. We demonstrate several properties of these networks, including noise suppression and amplification, irregular activity and synaptic balance, as well as how they relate to rate network dynamics in the limit that the boundary becomes soft. Finally, while our work focuses on small networks (5-50 neurons), we discuss potential avenues for scaling up to much larger networks. Overall, our work proposes a new perspective on spiking networks that may serve as a starting point for a mechanistic understanding of biological spike-based computation.
A new loss function for speaker recognition with deep neural network is proposed, based on Jeffreys Divergence. Adding this divergence to the cross-entropy loss function allows to maximize the target value of the output distribution while smoothing the non-target values. This objective function provides highly discriminative features. Beyond this effect, we propose a theoretical justification of its effectiveness and try to understand how this loss function affects the model, in particular the impact on dataset types (i.e. in-domain or out-of-domain w.r.t the training corpus). Our experiments show that Jeffreys loss consistently outperforms the state-of-the-art for speaker recognition, especially on out-of-domain data, and helps limit false alarms.
We propose to enhance the training of physics-informed neural networks (PINNs). To this aim, we introduce nonlinear additive and multiplicative preconditioning strategies for the widely used L-BFGS optimizer. The nonlinear preconditioners are constructed by utilizing the Schwarz domain-decomposition framework, where the parameters of the network are decomposed in a layer-wise manner. Through a series of numerical experiments, we demonstrate that both, additive and multiplicative preconditioners significantly improve the convergence of the standard L-BFGS optimizer, while providing more accurate solutions of the underlying partial differential equations. Moreover, the additive preconditioner is inherently parallel, thus giving rise to a novel approach to model parallelism.
In this note, we present an abstract approach to study asymptotic orders for adaptive approximations with respect to a monotone set function $\mathfrak{J}$ defined on dyadic cubes. We determine the exact upper order in terms of the critical value of the corresponding $\mathfrak{J}$-partition function, and we are able to provide upper and lower bounds in term of fractal-geometric quantities. With properly chosen $\mathfrak{J}$, our new approach has applications in many different areas of mathematics, including the spectral theory of Krein-Feller operators, quantization dimensions of compactly supported probability measures, and the exact asymptotic order for Kolmogorov, Gelfand and linear widths for Sobolev embeddings into $L_{\mu}^p$-spaces.
We consider the classical \emph{spherical} perceptrons and study their capacities. The famous zero-threshold case was solved in the sixties of the last century (see, \cite{Wendel62,Winder,Cover65}) through the high-dimensional combinatorial considerations. The general threshold, $\kappa$, case though turned out to be much harder and stayed out of reach for the following several decades. A substantial progress was then made in \cite{SchTir02} and \cite{StojnicGardGen13} where the \emph{positive} threshold ($\kappa\geq 0$) scenario was finally fully settled. While the negative counterpart ($\kappa\leq 0$) remained out of reach, \cite{StojnicGardGen13} did show that the random duality theory (RDT) is still powerful enough to provide excellent upper bounds. Moreover, in \cite{StojnicGardSphNeg13}, a \emph{partially lifted} RDT variant was considered and it was shown that the upper bounds of \cite{StojnicGardGen13} can be lowered. After recent breakthroughs in studying bilinearly indexed (bli) random processes in \cite{Stojnicsflgscompyx23,Stojnicnflgscompyx23}, \emph{fully lifted} random duality theory (fl RDT) was developed in \cite{Stojnicflrdt23}. We here first show that the \emph{negative spherical perceptrons} can be fitted into the frame of the fl RDT and then employ the whole fl RDT machinery to characterize the capacity. To be fully practically operational, the fl RDT requires a substantial numerical work. We, however, uncover remarkable closed form analytical relations among key lifting parameters. Such a discovery enables performing the needed numerical calculations to obtain concrete capacity values. We also observe that an excellent convergence (with the relative improvement $\sim 0.1\%$) is achieved already on the third (second non-trivial) level of the \emph{stationarized} full lifting.
Objective: This study aims to use artificial intelligence to realize the automatic planning of laminectomy, and verify the method. Methods: We propose a two-stage approach for automatic laminectomy cutting plane planning. The first stage was the identification of key points. 7 key points were manually marked on each CT image. The Spatial Pyramid Upsampling Network (SPU-Net) algorithm developed by us was used to accurately locate the 7 key points. In the second stage, based on the identification of key points, a personalized coordinate system was generated for each vertebra. Finally, the transverse and longitudinal cutting planes of laminectomy were generated under the coordinate system. The overall effect of planning was evaluated. Results: In the first stage, the average localization error of the SPU-Net algorithm for the seven key points was 0.65mm. In the second stage, a total of 320 transverse cutting planes and 640 longitudinal cutting planes were planned by the algorithm. Among them, the number of horizontal plane planning effects of grade A, B, and C were 318(99.38%), 1(0.31%), and 1(0.31%), respectively. The longitudinal planning effects of grade A, B, and C were 622(97.18%), 1(0.16%), and 17(2.66%), respectively. Conclusions: In this study, we propose a method for automatic surgical path planning of laminectomy based on the localization of key points in CT images. The results showed that the method achieved satisfactory results. More studies are needed to confirm the reliability of this approach in the future.
Recent years have seen increasing efforts to forecast infectious disease burdens, with a primary goal being to help public health workers make informed policy decisions. However, there has only been limited discussion of how predominant forecast evaluation metrics might indicate the success of policies based in part on those forecasts. We explore one possible tether between forecasts and policy: the allocation of limited medical resources so as to minimize unmet need. We use probabilistic forecasts of disease burden in each of several regions to determine optimal resource allocations, and then we score forecasts according to how much unmet need their associated allocations would have allowed. We illustrate with forecasts of COVID-19 hospitalizations in the US, and we find that the forecast skill ranking given by this allocation scoring rule can vary substantially from the ranking given by the weighted interval score. We see this as evidence that the allocation scoring rule detects forecast value that is missed by traditional accuracy measures and that the general strategy of designing scoring rules that are directly linked to policy performance is a promising direction for epidemic forecast evaluation.
In large-scale systems there are fundamental challenges when centralised techniques are used for task allocation. The number of interactions is limited by resource constraints such as on computation, storage, and network communication. We can increase scalability by implementing the system as a distributed task-allocation system, sharing tasks across many agents. However, this also increases the resource cost of communications and synchronisation, and is difficult to scale. In this paper we present four algorithms to solve these problems. The combination of these algorithms enable each agent to improve their task allocation strategy through reinforcement learning, while changing how much they explore the system in response to how optimal they believe their current strategy is, given their past experience. We focus on distributed agent systems where the agents' behaviours are constrained by resource usage limits, limiting agents to local rather than system-wide knowledge. We evaluate these algorithms in a simulated environment where agents are given a task composed of multiple subtasks that must be allocated to other agents with differing capabilities, to then carry out those tasks. We also simulate real-life system effects such as networking instability. Our solution is shown to solve the task allocation problem to 6.7% of the theoretical optimal within the system configurations considered. It provides 5x better performance recovery over no-knowledge retention approaches when system connectivity is impacted, and is tested against systems up to 100 agents with less than a 9% impact on the algorithms' performance.
We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.
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