Scalable method bounds Lipschitz constant of generative models.
problem Bounding the Lipschitz constant of generative models.
method Layerwise convex approximations using zonotopes.
result Efficient and tight bounds on generative models.
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13 results for “zonotopes”
Improves neural network verification by merging abstract domains and Lagrangian methods.
problem Prove provable bounds for neural network outputs given input ranges.
method Uses zonotopes within a Lagrangian decomposition to verify deep neural networks.
result Yields bounds that improve upon existing techniques in both time and tightness.
New polynomial-time solutions found for training ReLU networks, mirroring Max-Cut complexity.
problem Training two-layer ReLU neural networks with weight decay regularization.
method Developed a convex formulation and randomized algorithm to find approximate global optimizers.
result First polynomial-time approximation guarantees and hardness of approximation results for regularized ReLU networks.
This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to characterize the decision boundaries of a simple network of the form (Affine, Re…
We establish, for the first time, connections between feedforward neural networks with ReLU activation and tropical geometry --- we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden laye…
Convex optimization refines neural network training, improving model performance and reducing hyperparameter sensitivity.
problem Training deep neural networks using non-convex optimization methods often leads to suboptimal solutions and requires extensive tuning.
method Formulate neural network training as convex programs with regularization terms, leveraging sparse recovery models and semi-infinite programming theory.
result Convex models can achieve global optima and outperform traditional non-convex methods, with improved robustness to hyperparameters.
We derive relations between theoretical properties of restricted Boltzmann machines (RBMs), popular machine learning models which form the building blocks of deep learning models, and several natural notions from discrete mathematics and convex geometry. We give implications and equivalences relating RBM-representable …
The Wythoff construction takes a -dimensional polytope , a subset of and returns another -dimensional polytope . If is a regular polytope, then is vertex-transitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4. We want …
Shelling-type orderings of regular CW-complexes and acyclic matchings of the Salvetti complexmath.CO
Motivated by the work of Salvetti and Settepanella we introduce certain total orderings of the faces of any shellable regular CW-complex (called `shelling-type orderings') that can be used to explicitly construct maximum acyclic matchings of the poset of cells of the given complex. Building on an application of this me…
Determinantal point processes (DPPs) are distributions over sets of items that model diversity using kernels. Their applications in machine learning include summary extraction and recommendation systems. Yet, the cost of sampling from a DPP is prohibitive in large-scale applications, which has triggered an effort towar…
We study the problem of formally verifying the robustness to adversarial examples of support vector machines (SVMs), a major machine learning model for classification and regression tasks. Following a recent stream of works on formal robustness verification of (deep) neural networks, our approach relies on a sound abst…
Framework for safely updating machine learning models.
problem Continuous updates to machine learning models can lead to unintended consequences.
method Formalizes the problem as computing the largest locally invariant domain (LID), uses tractable primal-dual formulation.
result Matches or exceeds heuristic baselines for avoiding forgetting while providing formal safety guarantees.
In this paper we investigate the family of functions representable by deep neural networks (DNN) with rectified linear units (ReLU). We give an algorithm to train a ReLU DNN with one hidden layer to *global optimality* with runtime polynomial in the data size albeit exponential in the input dimension. Further, we impro…