In recent years, many deep learning approaches have incorporated layers that solve optimization problems (e.g., linear, quadratic, and semidefinite programs). Integrating these optimization problems as differentiable layers requires computing the derivatives of the optimization problem's solution with respect to its objective and constraints. This has so far prevented the use of state-of-the-art black-box numerical solvers within neural networks, as they lack a differentiable interface. To address this issue for one of the most common convex optimization problems -- quadratic programming (QP) -- we introduce dQP, a modular framework that enables plug-and-play differentiation for any QP solver, allowing seamless integration into neural networks and bi-level optimization tasks. Our solution is based on the core theoretical insight that knowledge of the active constraint set at the QP optimum allows for explicit differentiation. This insight reveals a unique relationship between the computation of the solution and its derivative, enabling efficient differentiation of any solver, that only requires the primal solution. Our implementation, which will be made publicly available, interfaces with an existing framework that supports over 15 state-of-the-art QP solvers, providing each with a fully differentiable backbone for immediate use as a differentiable layer in learning setups. To demonstrate the scalability and effectiveness of dQP, we evaluate it on a large benchmark dataset of QPs with varying structures. We compare dQP with existing differentiable QP methods, demonstrating its advantages across a range of problems, from challenging small and dense problems to large-scale sparse ones, including a novel bi-level geometry optimization problem.

This paper proposes a fully-automatic, text-guided generative method for producing periodic, repeating, tile-able 2D art, such as the one seen on floors, mosaics, ceramics, and the work of M.C. Escher. In contrast to the standard concept of a seamless texture, i.e., square images that are seamless when tiled, our method generates non-square tilings which comprise solely of repeating copies of the same object. It achieves this by optimizing both geometry and color of a 2D mesh, in order to generate a non-square tile in the shape and appearance of the desired object, with close to no additional background details. We enable geometric optimization of tilings by our key technical contribution: an unconstrained, differentiable parameterization of the space of all possible tileable shapes for a given symmetry group. Namely, we prove that modifying the laplacian used in a 2D mesh-mapping technique - Orbifold Tutte Embedding - can achieve all possible tiling configurations for a chosen planar symmetry group. We thus consider both the mesh's tile-shape and its texture as optimizable parameters, rendering the textured mesh via a differentiable renderer. We leverage a trained image diffusion model to define a loss on the resulting image, thereby updating the mesh's parameters based on its appearance matching the text prompt. We show our method is able to produce plausible, appealing results, with non-trivial tiles, for a variety of different periodic tiling patterns.

Exploring variations of 3D shapes is a time-consuming process in traditional 3D modeling tools. Deep generative models of 3D shapes often feature continuous latent spaces that can, in principle, be used to explore potential variations starting from a set of input shapes. In practice, doing so can be problematic: latent spaces are high dimensional and hard to visualize, contain shapes that are not relevant to the input shapes, and linear paths through them often lead to sub-optimal shape transitions. Furthermore, one would ideally be able to explore variations in the original high-quality meshes used to train the generative model, not its lower-quality output geometry. In this paper, we present a method to explore variations among a given set of landmark shapes by constructing a mapping from an easily-navigable 2D exploration space to a subspace of a pre-trained generative model. We first describe how to find a mapping that spans the set of input landmark shapes and exhibits smooth variations between them. We then show how to turn the variations in this subspace into deformation fields, to transfer those variations to high-quality meshes for the landmark shapes. Our results show that our method can produce visually-pleasing and easily-navigable 2D exploration spaces for several different shape categories, especially as compared to prior work on learning deformation spaces for 3D shapes.

The recent proliferation of 3D content that can be consumed on hand-held devices necessitates efficient tools for transmitting large geometric data, e.g., 3D meshes, over the Internet. Detailed high-resolution assets can pose a challenge to storage as well as transmission bandwidth, and level-of-detail techniques are often used to transmit an asset using an appropriate bandwidth budget. It is especially desirable for these methods to transmit data progressively, improving the quality of the geometry with more data. Our key insight is that the geometric details of 3D meshes often exhibit similar local patterns even across different shapes, and thus can be effectively represented with a shared learned generative space. We learn this space using a subdivision-based encoder-decoder architecture trained in advance on a large collection of surfaces. We further observe that additional residual features can be transmitted progressively between intermediate levels of subdivision that enable the client to control the tradeoff between bandwidth cost and quality of reconstruction, providing a neural progressive mesh representation. We evaluate our method on a diverse set of complex 3D shapes and demonstrate that it outperforms baselines in terms of compression ratio and reconstruction quality.

We propose an end-to-end deep-learning approach for automatic rigging and retargeting of 3D models of human faces in the wild. Our approach, called Neural Face Rigging (NFR), holds three key properties: (i) NFR's expression space maintains human-interpretable editing parameters for artistic controls; (ii) NFR is readily applicable to arbitrary facial meshes with different connectivity and expressions; (iii) NFR can encode and produce fine-grained details of complex expressions performed by arbitrary subjects. To the best of our knowledge, NFR is the first approach to provide realistic and controllable deformations of in-the-wild facial meshes, without the manual creation of blendshapes or correspondence. We design a deformation autoencoder and train it through a multi-dataset training scheme, which benefits from the unique advantages of two data sources: a linear 3DMM with interpretable control parameters as in FACS, and 4D captures of real faces with fine-grained details. Through various experiments, we show NFR's ability to automatically produce realistic and accurate facial deformations across a wide range of existing datasets as well as noisy facial scans in-the-wild, while providing artist-controlled, editable parameters.

Finding multiple solutions of non-convex optimization problems is a ubiquitous yet challenging task. Most past algorithms either apply single-solution optimization methods from multiple random initial guesses or search in the vicinity of found solutions using ad hoc heuristics. We present an end-to-end method to learn the proximal operator of a family of training problems so that multiple local minima can be quickly obtained from initial guesses by iterating the learned operator, emulating the proximal-point algorithm that has fast convergence. The learned proximal operator can be further generalized to recover multiple optima for unseen problems at test time, enabling applications such as object detection. The key ingredient in our formulation is a proximal regularization term, which elevates the convexity of our training loss: by applying recent theoretical results, we show that for weakly-convex objectives with Lipschitz gradients, training of the proximal operator converges globally with a practical degree of over-parameterization. We further present an exhaustive benchmark for multi-solution optimization to demonstrate the effectiveness of our method. Searching for multiple optima of an optimization problem is a ubiquitous yet under-explored task. In applications like low-rank recovery (Ge et al., 2017), topology optimization (Papadopoulos et al., 2021), object detection (Lin et al., 2014), and symmetry detection (Shi et al., 2020), it is desirable to recover multiple near-optimal solutions, either because there are many equally-performant global optima or due to the fact that the optimization objective does not capture user preferences precisely. Even for single-solution non-convex optimization, typical methods look for multiple local optima from random initial guesses before picking the best local optimum. Additionally, it is often desirable to obtain solutions to a family of optimization problems with parameters not known in advance, for instance, the weight of a regularization term, without having to restart from scratch. R is the objective function depending on τ. The goal of MSO is to identify multiple solutions for each τ T, i.e., the set {x As finding all global minima in the general case is extremely challenging, realistically our goal is to find a diverse set of local minima.