Paper proposes a new method to improve subspace clustering by using a tighter rank approximation.
problem Challenges in subspace clustering due to the limitations of the nuclear norm approximation.
method Uses an arctangent function as a tighter approximation to the rank function and develops an optimization procedure based on ALM.
result Demonstrates effectiveness of the proposed method through experiments on face clustering and motion segmentation.
New algorithms separate singing voices from accompaniment using complex and quaternionic principal component pursuit.
problem Separating singing voices from instrumental accompaniment using phase information.
method Extended principal component pursuit to complex and quaternionic cases, developed new proximity operators, applied inexact augmented Lagrange multiplier algorithm.
result Phase information improves singing voice separation.
The paper proposes using LogDet for better low-rank approximation in subspace clustering.
problem Improving low-rank approximation for better clustering performance.
method Using LogDet as a non-convex approximation to rank, optimizing with augmented Lagrange multipliers.
result The proposed method often outperforms state-of-the-art subspace clustering algorithms.
A new method for robust PCA using nonconvex rank approximation.
problem Recovering a matrix of minimal rank in data mining and machine learning.
method Proposes a nonconvex rank approximation to the nuclear norm, solving the associated nonconvex minimization problem with an efficient algorithm.
result Our method outperforms current state-of-the-art algorithms in both accuracy and efficiency.
Researchers use Gaussian processes to approximate Lagrange multipliers for Maximum-Entropy distributions.
problem Finding Lagrange multipliers for Maximum-Entropy distributions is computationally challenging.
method Employed Gaussian processes to approximate the Lagrange multipliers as a map of moments. Optimized hyperparameters by maximizing log-likelihood.
result Data-driven Maximum-Entropy closure performs well in approximating non-equilibrium distributions.
Paper improves MVSC using tensor low-rank modeling.
problem Improving multi-view spectral clustering.
method Structured tensor low-rank norm for MVSC optimization.
result Proposed method outperforms state-of-the-art methods.
The paper explores the correspondence between gradient flow lines of a function and its Lagrange multiplier functional.
problem Detecting critical points of a function subject to constraints.
method Adiabatic limit technique and singular version of the implicit function theorem.
result A one-to-one correspondence between gradient flow lines connecting critical points of Morse index difference one.
Enhances ordinal embedding with less data by focusing on margin distribution.
problem Insufficient labeled data for ordinal embedding.
method Proposes Distributional Margin based Ordinal Embedding (DMOE) to improve generalization with less data.
result Demonstrates improved generalization performance with less labeled data.
Stochastic approach improves neural network training for kinetic simulations.
problem Training neural networks under physical constraints in kinetic fusion simulations.
method Stochastic augmented Lagrangian approach using pyTorch.
result Higher model prediction accuracy achieved compared to fixed penalty method.
Optimization with inequality constraints using embedded gradient vector field method
problem Optimization with inequality constraints
method Geometric framework using quadratic slack variables
result Derives Lagrange multiplier functions and second-order optimality conditions
It is shown that the Euler-Lagrange equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of reduction and the relation with Lagrange multiplier method are also studied.
Discrete Lagrange problems solved with Lie group constraints.
problem Solving discrete Lagrange problems with Lie group constraints.
method Proving critical sections are solutions of unconstrained variational problems, applying Noether theory and multisymplectic forms.
result Critical sections of discrete Lagrange problems are solutions of unconstrained variational problems.
Agents learn shared dictionary elements and parameters in a decentralized online setting.
problem Discriminative dictionary learning in a distributed online setting.
method Formulated as a distributed stochastic program, solved using a block variant of the Arrow-Hurwicz saddle point algorithm with Lagrange multipliers.
result Decisions asymptotically achieve a first-order stationarity condition on average.
Researchers prove existence of smooth hypersurface in hyperbolic space.
problem Existence of a smooth complete 3-convex hypersurface in hyperbolic space.
method Lagrange multiplier method to compute extreme value of concavity.
result Existence of a smooth complete 3-convex hypersurface satisfying curvature equation and asymptotic boundary.
A new algorithm finds optimal solutions for constrained decision processes.
problem Optimizing state-value functions with constraints in CMDPs.
method Gradient-Aware Search (GAS) exploiting PWLC structure.
result GAS converges faster and more reliably than existing methods.
A new method solves variational inequality problems with multiple constraints without needing optimal Lagrange multipliers.
problem Solving variational inequality problems with multiple functional constraints efficiently.
method Constrained Gradient Method (CGM) for Minty variational inequality problems.
result The Constrained Gradient Method achieves complexity similar to projection-based methods but with cheaper oracles.
Paper analyzes portfolio optimization problems using random matrix approach.
problem Minimizing/maximizing investment risk and concentration under identical variances.
method Lagrange multiplier method and random matrix approach.
result Validation of results through numerical experiments.
Maximizes capacity of extensions with fixed boundary data.
problem Maximizing the capacity of extensions with nonnegative scalar curvature.
method Using the method of Lagrange multipliers on the constraint space of scalar-flat extensions.
result Derives variational condition for maximal capacity extensions and proves they have constant scalar curvature.
New algorithm reduces Q-value overestimation in deep reinforcement learning.
problem Q-value overestimation in deep reinforcement learning for high-dimensional state spaces.
method Adapting information theory concepts, an intrinsic penalty signal is introduced to encourage reduced Q-value estimates.
result Algorithm outperforms other methods in Atari games, demonstrating efficient and robust learning.
Given two Morse functions f,μ on a compact manifold M, we study the Morse homology for the Lagrange multiplier function on M×R which sends (x,η) to f(x)+ημ(x). Take a product metric on M×R, and rescale its R-component by a factor λ2. We show that generica…
Simplifies neural network models by explicitly enforcing constraints in Cartesian coordinates.
problem Learning dynamics of complex systems efficiently and accurately.
method Embedding systems into Cartesian coordinates and using Lagrange multipliers to enforce constraints.
result Explicitly enforcing constraints leads to a 100x improvement in accuracy and data efficiency.
A continuous-time financial portfolio selection model with expected utility maximization typically boils down to solving a (static) convex stochastic optimization problem in terms of the terminal wealth, with a budget constraint. In literature the latter is solved by assuming {\it a priori} that the problem is well-pos…
Extending Lévi-Civita's concept to non-quadratic spaces, this study finds extremal compatible linear connections.
problem Extending the Lévi-Civita connection to non-quadratic spaces.
method Hybrid conditional extremum problem, Lagrange multipliers, geometric approach.
result Existence and characterization of extremal compatible linear connections.
The paper studies Einstein-Hilbert action on complex manifolds.
problem Deriving equations for the Einstein-Hilbert action on almost k-product manifolds. method Adapted variations of metric, deriving Euler-Lagrange equations.
result Presented a nice form of Einstein equation.
The paper shows how various generative models are related through a unified optimization problem.
problem Training latent variable generative models with different objectives.
method Unified Lagrangian perspective, optimizing mutual information and constraints.
result Unified characterization of training objectives and Pareto optimal solutions.
Deep learning technique solves large 0-1 knapsack problems quickly.
problem Large-scale 0-1 knapsack problems with variable item values.
method Adaptive gradient ascent method based on Lagrange multiplier and game theory.
result Solves all large-scale benchmark KP instances in a minute.
Study S-shaped utility maximization with VaR constraint and unobservable drift.
problem Maximizing utility with a Value at Risk (VaR) constraint and unknown drift.
method Bayesian filter, concavification principle, change of measure, semi-closed integral representation, algorithms (Lagrange, simulation, deep neural network).
result Critical wealth level determining solution feasibility and optimal solution existence.
Bayesian optimization tackles mixed discrete-continuous problems with Gaussian processes.
problem Optimizing problems with both discrete and continuous variables using costly simulations.
method Relaxing discrete variables into continuous latent variables, using Bayesian optimization, and incorporating compatibility constraints with Lagrangians.
result Comparative analysis of different mixed Bayesian optimization approaches.
In this paper we provide generalized Helmholtz conditions, in terms of a semi-basic 1-form, which characterize when a given system of second order ordinary differential equations is equivalent to the Lagrange equations, for some given arbitrary non-conservative forces. For the particular cases of dissipative or gyrosco…
Study on membranes under confinement, proving existence and regularity of minimizers.
problem Existence and regularity of minimizers for constrained Helfrich energy.
method Elliptic system analysis, careful study of measure-valued Lagrange multiplier.
result Optimal regularity for solutions throughout branch points, rigid behavior for unit ball minimizers.
We consider the classical optimal dividends problem under the Cramér-Lundberg model with exponential claim sizes subject to a constraint on the time of ruin. We introduce the dual problem and show that the complementary slackness conditions are satisfied, thus there is no duality gap. Therefore the optimal value functi…
Fenchel lifted networks improve neural network training without performance loss.
problem The difficulty and non-convexity of training deep neural networks.
method Introduces Fenchel lifted networks that represent activation functions as biconvex constraints and uses Lagrange Multipliers to create a lower bound of the training problem.
result Fenchel lifted networks match or outperform traditional neural networks in performance.
Develops Frank-Wolfe Augmented Lagrangian for convex optimization.
problem Minimizing functions over intersections of convex sets.
method Frank-Wolfe Augmented Lagrangian (FW-AL) method.
result Sublinear convergence rate for general convex compact sets, linear for polytopes.
Study of critical points for 4D conformally invariant curvature energies.
problem Analyzing critical points of conformally invariant curvature energies in 4 dimensions.
method Using Noether's theorem and divergence-free potentials, generating an algebraic structure, and considering Palais-Smale sequences.
result Improved energy estimates for critical points under small-energy hypotheses.
Unified framework for network model assessment using maximum entropy.
problem Statistical inference for network models.
method Constrained entropy-maximization problem, Lagrange multipliers.
result Consistent goodness-of-fit and two-sample tests for network models.
Solves an Arnold trivium problem using calculus and topology.
problem Finding critical points on a two-dimensional surface.
method Lagrange multipliers, Morse theory, Poincare-Hopf theorem.
result Determines the genus of a two-dimensional surface.
Study derives a limit functional for Willmore graphs with curvature penalization.
problem Optimizing Willmore graphs with curvature constraints.
method Interpreting penalization as Lagrange multiplier, deriving Γ-limit. result Derives a new limit functional for Willmore graphs.
Critical trajectories in a sphere are found for a specific bending functional.
problem Finding closed trajectories in a sphere for a specific bending functional.
method Existence of infinitely many closed trajectories shown for a given Lagrange multiplier.
result Existence of closed trajectories dependent on a pair of relatively prime natural numbers.
We study a new bordification of the decorated Teichmüller space for a multiply punctured surface F by a space of filtered screens on the surface that arises from a natural elaboration of earlier work of McShane-Penner. We identify necessary and sufficient conditions for paths in this space of filtered screens to yield …
We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibere…
Global implicit function theorem for Fréchet spaces, solving derivative loss problems.
problem Solving initial value problems with derivative loss in Fréchet spaces.
method Global implicit function theorems for Keller's Cc1-mappings in Fréchet spaces, applied through submersions and transversality. result Global existence and uniqueness of solutions to initial value problems with derivative loss.
The paper proves transversality for special Lagrangian submanifolds in a 6D manifold.
problem Counting special Lagrangian submanifolds in higher dimensions.
method Proving transversality for the moduli space of perturbed special Lagrangian submanifolds using a Lagrange multipliers problem.
result The moduli space is generically a set of isolated points.
We investigate the geometry of hyperbolic knots and links whose diagrams have a high amount of twisting of multiple strands. We find information on volume and certain isotopy classes of geodesics for the complements of these links, based only on a diagram. The results are obtained by finding geometric information on ge…
This paper solves a utility maximization problem under utility-based shortfall risk constraint, by proposing an approach using Lagrange multiplier and convex duality. Under mild conditions on the asymptotic elasticity of the utility function and the loss function, we find an optimal wealth process for the constrained p…
We study the problem of large scale, multi-label visual recognition with a large number of possible classes. We propose a method for augmenting a trained neural network classifier with auxiliary capacity in a manner designed to significantly improve upon an already well-performing model, while minimally impacting its c…
In this paper we consider a Lagrange Multiplier-type test (LM) to detect change in the mean of time series with heteroskedasticity of unknown form. We derive the limiting distribution under the null, and prove the consistency of the test against the alternative of either an abrupt or smooth changes in the mean. We perf…
This paper introduces a new Lagrangian for the Information Bottleneck problem to simplify optimization.
problem Optimizing compressed representations for predicting Y while limiting information about X. method Introduces a general family of Lagrangians to explore the Information Bottleneck curve.
result Solves the original constrained optimization problem with a single optimization.
Method solves nonconvex constrained optimization problems with a new augmented Lagrangian approach.
problem Nonconvex composite functional constraints with inequality constraints.
method First-order augmented Lagrangian method with smoothed prox-linear reformulation.
result Explicit convergence rates for the proposed method in terms of KKT residual.