Additionally, we show generalization performance of our proposed method across differentvisualdomains. Withthegiven problemcategory(task),asubsetforlearning can be sampled (via domain episode module in Figure 4 in main text). Here, by replacingclass with task, K-shot andN-task reasoning framework can be defined. Here, we show analogical learning with the existing meta learning framework for fast adaptation fromthesourcedomain tothetargetdomain.

However, considering the ever-evolving paradigms in deep learning, employees with ulterior motivesmay fabricate reasons such asthe requirements ofdata augmentation [6]orthe purpose of multimodal learning [3] to apply for relevant and irrelevant private datasets, which is common in social engineering [4].

In order apply the change-of-variables formula to get a density for the generator, we assume that G: Rd RD spans an immersedd-dimensional manifold inRD. The governing assumption is that the Jacobian ofG exist and has full rank. However, we note that one requirement is that no hidden layer may have dimensionality below the d dimensions of the latent space. This is a natural requirement for the generator anyway. In our model, we aim to maximize the entropy of the generator, which encourages the generator to create as diverse samples as possible.

In the limitβ, the measureµβ concentrates around solutions of the minimisation in eq.(24).

The current version of Predify assumes that there is no gap between the encoders. One can easily override the default setting by providing all the details for a PCoder. A.3 ExecutionTime Since we used a variable number of GPUs for the different experiments, an exact execution time is hard to pinpoint. We expect that this could be further improved with a more extensive and systematic hyperparameter search. In other words, their training hyperparameters appeared to have been optimised for their predictive coding network, but not - or not as much - for their feedforward baseline.

In what follows we will call each experiment by its corresponding figure or table number for convenience. For the rotated/shifted MNIST images (Figure 8, 9), we use the Affine transformation function in the TorchVisionlibrary. In experiments (Table 2, 3, 4, 5), we use either or both of the Large (L) and Small (S) dataset for the standard benchmark vision data: MNIST, FMNIST, KMNIST, Omniglot, SVHN, CIFAR10, CIFAR100, CELEBA. For Figure 10, Table 3, the regularization coefficients for CAE, WAE are searched around 0.01 0.001, the noise level used in DAE is searched around0.1 0.01, and the regularization coefficient andλforSPAEandNRAE aresearched around0.001 Ontheother hand, the runtimes of our algorithms are comparable with other existing methods.

Mean-field games (MFGs) study the Nash equilibrium of systems with a continuum of interacting agents, which can be formulated as the fixed-point of optimal control problems. They provide a unified framework for a variety of applications, including optimal transport (OT) and generative models. Despite their broad applicability, solving high-dimensional MFGs remains a significant challenge due to fundamental computational and analytical obstacles. In this work, we propose a particle-based deep Flow Matching (FM) method to tackle high-dimensional MFG computation. In each iteration of our proximal fixed-point scheme, particles are updated using first-order information, and a flow neural network is trained to match the velocity of the sample trajectories in a simulation-free manner. Theoretically, in the optimal control setting, we prove that our scheme converges to a stationary point sublinearly, and upgrade to linear (exponential) convergence under additional convexity assumptions. Our proof uses FM to induce an Eulerian coordinate (density-based) from a Lagrangian one (particle-based), and this also leads to certain equivalence results between the two formulations for MFGs when the Eulerian solution is sufficiently regular. Our method demonstrates promising performance on non-potential MFGs and high-dimensional OT problems cast as MFGs through a relaxed terminal-cost formulation.

For Table 2, we use the large dataset. For Table 4 and 5, we use the small dataset. In this paper, we use fully connected neural network and convolutional neural network. For W AE, we use the MMD loss and median heuristic. For Figure 4, we use the networks of size (2-1024-1024-1) and (1-1024-1024-2) with ReLU activation functions for the encoder and decoder, respectively.