Space complexity in polynomial calculus

During the last decade, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused on weak systems that are used by SAT solvers. There has been a relatively long sequence of papers on space in resolution, which is now reasonably well understood from this point of view. For other natural candidates to study, however, such as polynomial calculus or cutting planes, very little has been known. We are not aware of any nontrivial space lower bounds for cutting planes, and for polynomial calculus the only lower bound has been for CNF formulas of unbounded width in [Alekhnovich et al. ’02], where the space lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could be able to refute any k-CNF formula in constant space. In this paper, we prove several new results on space in polynomial calculus (PC), and in the extended proof system polynomial calculus resolution (PCR) studied in [Alekhnovich et al. ’02]: 1. We prove an Ω(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole principle formulas PHPm n with m pigeons and n holes, and show that this is tight. 2. For PCR, we prove an Ω(n) space lower bound for a bitwise encoding of the functional pigeonhole principle. These formulas have width O(log n), and hence this is an exponential improvement over [Alekhnovich et al. ’02] measured in the width of the formulas. 3. We then present another encoding of the pigeonhole principle that has constant width, and prove an Ω(n) space lower bound in PCR for these formulas as well. 4. Finally, we prove that any k-CNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not obviously the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way, something that we believe can be useful when proving PCR space lower bounds for other well-studied formula families in proof complexity

Structured Learning of Tree Potentials in CRF for Image Segmentation

We propose a new approach to image segmentation, which exploits the advantages of both conditional random fields (CRFs) and decision trees. In the literature, the potential functions of CRFs are mostly defined as a linear combination of some pre-defined parametric models, and then methods like structured support vector machines (SSVMs) are applied to learn those linear coefficients. We instead formulate the unary and pairwise potentials as nonparametric forests---ensembles of decision trees, and learn the ensemble parameters and the trees in a unified optimization problem within the large-margin framework. In this fashion, we easily achieve nonlinear learning of potential functions on both unary and pairwise terms in CRFs. Moreover, we learn class-wise decision trees for each object that appears in the image. Due to the rich structure and flexibility of decision trees, our approach is powerful in modelling complex data likelihoods and label relationships. The resulting optimization problem is very challenging because it can have exponentially many variables and constraints. We show that this challenging optimization can be efficiently solved by combining a modified column generation and cutting-planes techniques. Experimental results on both binary (Graz-02, Weizmann horse, Oxford flower) and multi-class (MSRC-21, PASCAL VOC 2012) segmentation datasets demonstrate the power of the learned nonlinear nonparametric potentials.Comment: 10 pages. Appearing in IEEE Transactions on Neural Networks and Learning System

The Many Moods of Emotion

This paper presents a novel approach to the facial expression generation problem. Building upon the assumption of the psychological community that emotion is intrinsically continuous, we first design our own continuous emotion representation with a 3-dimensional latent space issued from a neural network trained on discrete emotion classification. The so-obtained representation can be used to annotate large in the wild datasets and later used to trained a Generative Adversarial Network. We first show that our model is able to map back to discrete emotion classes with a objectively and subjectively better quality of the images than usual discrete approaches. But also that we are able to pave the larger space of possible facial expressions, generating the many moods of emotion. Moreover, two axis in this space may be found to generate similar expression changes as in traditional continuous representations such as arousal-valence. Finally we show from visual interpretation, that the third remaining dimension is highly related to the well-known dominance dimension from psychology

On the Power and Limitations of Branch and Cut

The Stabbing Planes proof system [Paul Beame et al., 2018] was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas - certain unsatisfiable systems of linear equations od 2 - which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari [Daniel Dadush and Samarth Tiwari, 2020] showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes. In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients (SP*) can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas

Understanding Cutting Planes for QBFs

We define a cutting planes system CP+8red for quantified Boolean formulas (QBF) and analyse the proof-theoretic strength of this new calculus. While in the propositional case, Cutting Planes is of intermediate strength between resolution and Frege, our findings here show that the situation in QBF is slightly more complex: while CP+8red is again weaker than QBF Frege and stronger than the CDCL-based QBF resolution systems Q-Res and QU-Res, it turns out to be incomparable to even the weakest expansion-based QBF resolution system 8Exp+Res. Technically, our results establish the effectiveness of two lower boun