Face parsing is a fundamental task in computer vision, enabling applications such as identity verification, facial editing, and controllable image synthesis. However, existing face parsing models often lack fairness and robustness, leading to biased segmentation across demographic groups and errors under occlusions, noise, and domain shifts. These limitations affect downstream face synthesis, where segmentation biases can degrade generative model outputs. We propose a multi-objective learning framework that optimizes accuracy, fairness, and robustness in face parsing. Our approach introduces a homotopy-based loss function that dynamically adjusts the importance of these objectives during training. To evaluate its impact, we compare multi-objective and single-objective U-Net models in a GAN-based face synthesis pipeline (Pix2PixHD). Our results show that fairness-aware and robust segmentation improves photorealism and consistency in face generation. Additionally, we conduct preliminary experiments using ControlNet, a structured conditioning model for diffusion-based synthesis, to explore how segmentation quality influences guided image generation. Our findings demonstrate that multi-objective face parsing improves demographic consistency and robustness, leading to higher-quality GAN-based synthesis.11Source code and trained model weights are available on GitHub: https://github.com/sabraha2/Towards-Fair-and-Robust-Face-Parsing-for-Generative-AI-A-Multi-Objective-Approach.

The configuration spaces of parallel manipulators exhibit more nonlinearity than serial manipulators. Qualitatively, they can be seen to possess extra folds. Projection onto smaller spaces of engineering relevance, such as an output workspace or an input actuator space, these folds cast edges that exhibit boundary behavior. For example, inside the global workspace bounds of a five-bar linkage appear several local workspace bounds that only constrain certain output modes of the mechanism. The presence of such boundaries, which manifest in both input and output projections, serve as a source of confusion when these projections are studied exclusively instead of the configuration space itself. Particularly, the design of nonsymmetric parallel manipulators has been confounded by the presence of exotic projections in their input and output spaces. In this paper, we represent the configuration space with a radius graph, then weight each edge by solving an optimization problem using homotopy continuation to quantify transmission quality. We then employ a graph path planner to approximate geodesics between configuration points that avoid regions of low transmission quality. Our methodology automatically generates paths capable of transitioning between non-neighboring output modes, a motion which involves osculating multiple workspace boundaries (local, global, or both). We apply our technique to two nonsymmetric five-bar examples that demonstrate how transmission properties and other characteristics of the workspace can be selected by switching output modes.

We present a method for solving two minimal problems for relative camera pose estimation from three views, which are based on three view correspondences of (i) three points and one line and the novel case of (ii) three points and two lines through two of the points. These problems are too difficult to be efficiently solved by the state of the art Gröbner basis methods. Our method is based on a new efficient homotopy continuation (HC) solver framework MINUS, which dramatically speeds up previous HC solving by specializing hc methods to generic cases of our problems. We characterize their number of solutions and show with simulated experiments that our solvers are numerically robust and stable under image noise, a key contribution given the borderline intractable degree of nonlinearity of trinocular constraints. We show in real experiments that (i) sift feature location and orientation provide good enough point-and-line correspondences for three-view reconstruction and (ii) that we can solve difficult cases with too few or too noisy tentative matches, where the state of the art structure from motion initialization fails.

By using the viewpoint of modern computational algebraic geometry, we explore properties of the optimization landscapes of deep linear neural network models. After providing clarification on the various definitions of “flat” minima, we show that the geometrically flat minima, which are merely artifacts of residual continuous symmetries of the deep linear networks, can be straightforwardly removed by a generalized$L_2$-regularization. Then, we establish upper bounds on the number of isolated stationary points of these networks with the help of algebraic geometry. Combining these upper bounds with a method in numerical algebraic geometry, we findallstationary points for modest depth and matrix size. We demonstrate that, in the presence of the non-zero regularization, deep linear networks can indeed possess local minima which are not global minima. Finally, we show that even though the number of stationary points increases as the number of neurons (regularization parameters) increases (decreases), higher index saddles are surprisingly rare.

Polar linkages have two degrees-of-freedom (DOF) where one input joint angle controls the length of a radial segment while another controls its angle. Considering a theoretical planar robot model, this mapping between joint angles to output motions can be shown to be energetically advantageous over the ubiquitous two-revolute linkage. Since a polar linkage’s typical construction involves a moving prismatic joint, it is cumbersome to implement alongside rotary electromagnetic actuators offsetting any advantage. In this paper, we present a procedure for designing polar linkages using only revolute joints. The procedure starts with a pre-existing single DOF straight line linkage and then finds the dimensions of a three-link attachment to produce the second DOF. In the end, the straight line linkage actuates the polar length and the attachment actuates the polar angle. The design process is framed under optimization with an objective that is both polynomial and invariant to the number of discretization points. This enables the techniques of numerical continuation to efficiently find complete sets of minima. We demonstrate our procedure with an example in which multiple minima are found including the global minimum. This computed design solution is then fabricated in order to validate the designed kinematics.

We present a method for solving two minimal problems for relative camera pose estimation from three views, which are based on three view correspondences of (i) three points and one line and (ii) three points and two lines through two of the points. These problems are too difficult to be efficiently solved by the state of the art Grobner basis methods. Our method is based on a new efficient homotopy continuation (HC) solver, which dramatically speeds up previous HC solving by specializing HC methods to generic cases of our problems. We show in simulated experiments that our solvers are numerically robust and stable under image noise. We show in real experiment that (i) SIFT features provide good enough point-and-line correspondences for three-view reconstruction and (ii) that we can solve difficult cases with too few or too noisy tentative matches where the state of the art structure from motion initialization fails.

Topological data analysis (TDA) provides tools for computing geometric and topological information about spaces from a finite sample of points. We present an adaptive algorithm for finding provably dense samples of points on real algebraic varieties given a set of defining polynomials for use as input to TDA. The algorithm utilizes methods from numerical algebraic geometry to give formal guarantees about the density of the sampling, and also employs geometric heuristics to reduce the size of the sample. As TDA methods consume significant computational resources that scale poorly in the number of sample points, our sampling minimization makes applying TDA methods more feasible. We provide a software package that implements the algorithm, and showcase it through several examples.

Bertini_real is a compiled command line program for numerically decomposing the real portion of a positive-dimensional complex component of an algebraic set. The software uses homotopy continuation to solve a series of systems via regeneration from a witness set to compute a cell decomposition. The implemented decomposition algorithms are similar to the well-known cylindrical algebraic decomposition (CAD) first established by Collins in that they produce a set of connected cells. In contrast to the CAD, Bertini_real produces cells with midpoints connected to boundary points by homotopies, which can easily be numerically tracked. Furthermore, the implemented decomposition for surfaces naturally yields a triangulation. This CAD-like decomposition captures the topological information and permits further computation on the real sets, such as sampling, visualization, and three-dimensional printing.

Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares programming to test whether a given set is "complete" with respect to the real solution set. Specifically, we test whether the Zariski closure of that set is indeed equal to the solution set of the real radical of the ideal generated by the given polynomials. Examples with finitely and infinitely many real solutions are provided, along with an example having polynomial inequalities.

This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This con- struction uses a single linear differential form defined from the Jacobian matrix of the input, and defines the deflated system by applying this differential form to the original system. The advantages of this new deflation is that it does not introduce new variables and the increase in the number of equations is linear instead of the quadratic increase of previous methods. The second construction gives the coefficients of the so-called inverse system or dual basis, which defines the multiplicity structure at the singular root. We present a system of equations in the original variables plus a relatively small number of new variables. We show that the roots of this new system include the original singular root but now with multiplicity one, and the new variables uniquely determine the multiplicity structure. Both constructions are 'exact' in that they permit one to treat all conjugate roots simultaneously and can be used in certification procedures for singular roots and their multiplicity structure with respect to an exact rational polynomial system.

A Newton homotopy is a homotopy that involves changing only the constant terms. They arise naturally, for example, when performing monodromy loops, moving end effectors of robots, and simply when trying to compute a solution to a square system of equations. Previous certified path tracking techniques have focused on using an a priori certified tracking scheme which means that the stepsize is constructed so that the result automatically satisfies some conditions. These schemes use pessimistic stepsizes that can be much smaller than those used by heuristic tracking methods. This article designs an a posteriori certification scheme that uses the result of a heuristic tracking scheme as input to produce a certificate that the path was indeed tracked correctly, e.g., no path jumpings occurred. By using an a posteriori approach, each step can be certified independently and thus certification of the path can be performed in parallel. Examples are presented demonstrating the efficiency of this a posteriori certification approach.

Smale’s α -theory uses estimates related to the convergence of Newton’s method to certify that Newton iterations will converge quadratically to solutions to a square polynomial system. The program alphaCertified implements algorithms based on α -theory to certify solutions of polynomial systems using both exact rational arithmetic and arbitrary precision floating point arithmetic. It also implements algorithms that certify whether a given point corresponds to a real solution, and algorithms to heuristically validate solutions to overdetermined systems. Examples are presented to demonstrate the algorithms.