Bounding Embeddings of VC Classes into Maximum Classes

One of the earliest conjectures in computational learning theory-the Sample Compression conjecture-asserts that concept classes (equivalently set systems) admit compression schemes of size linear in their VC dimension. To-date this statement is known to be true for maximum classes---those that possess maximum cardinality for their VC dimension. The most promising approach to positively resolving the conjecture is by embedding general VC classes into maximum classes without super-linear increase to their VC dimensions, as such embeddings would extend the known compression schemes to all VC classes. We show that maximum classes can be characterised by a local-connectivity property of the graph obtained by viewing the class as a cubical complex. This geometric characterisation of maximum VC classes is applied to prove a negative embedding result which demonstrates VC-d classes that cannot be embedded in any maximum class of VC dimension lower than 2d. On the other hand, we show that every VC-d class C embeds in a VC-(d+D) maximum class where D is the deficiency of C, i.e., the difference between the cardinalities of a maximum VC-d class and of C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible results on embedding into maximum classes. For some special classes of Boolean functions, relationships with maximum classes are investigated. Finally we give a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum classes for smallest k.Comment: 22 pages, 2 figure

Quadrilateral-octagon coordinates for almost normal surfaces

Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension of this vector space from 7n to 3n (where n is the complexity of the underlying triangulation). Here we develop an analogous theory for octagonal almost normal surfaces, using quadrilateral and octagon coordinates to reduce this dimension from 10n to 6n. As an application, we show that quadrilateral-octagon coordinates can be used exclusively in the streamlined 3-sphere recognition algorithm of Jaco, Rubinstein and Thompson, reducing experimental running times by factors of thousands. We also introduce joint coordinates, a system with only 3n dimensions for octagonal almost normal surfaces that has appealing geometric properties.Comment: 34 pages, 20 figures; v2: Simplified the proof of Theorem 4.5 using cohomology, plus other minor changes; v3: Minor housekeepin

Computational topology with Regina: Algorithms, heuristics and implementations

Regina is a software package for studying 3-manifold triangulations and normal surfaces. It includes a graphical user interface and Python bindings, and also supports angle structures, census enumeration, combinatorial recognition of triangulations, and high-level functions such as 3-sphere recognition, unknot recognition and connected sum decomposition. This paper brings 3-manifold topologists up-to-date with Regina as it appears today, and documents for the first time in the literature some of the key algorithms, heuristics and implementations that are central to Regina's performance. These include the all-important simplification heuristics, key choices of data structures and algorithms to alleviate bottlenecks in normal surface enumeration, modern implementations of 3-sphere recognition and connected sum decomposition, and more. We also give some historical background for the project, including the key role played by Rubinstein in its genesis 15 years ago, and discuss current directions for future development.Comment: 29 pages, 10 figures; v2: minor revisions. To appear in "Geometry & Topology Down Under", Contemporary Mathematics, AM

Near-Optimal Evasion of Convex-Inducing Classifiers

Classifiers are often used to detect miscreant activities. We study how an adversary can efficiently query a classifier to elicit information that allows the adversary to evade detection at near-minimal cost. We generalize results of Lowd and Meek (2005) to convex-inducing classifiers. We present algorithms that construct undetected instances of near-minimal cost using only polynomially many queries in the dimension of the space and without reverse engineering the decision boundary.Comment: 8 pages; to appear at AISTATS'201

Exploiting Machine Learning to Subvert Your Spam Filter

Using statistical machine learning for making security decisions introduces new vulnerabilities in large scale systems. This paper shows how an adversary can exploit statistical machine learning, as used in the SpamBayes spam filter, to render it useless—even if the adversary’s access is limited to only 1 % of the training messages. We further demonstrate a new class of focused attacks that successfully prevent victims from receiving specific email messages. Finally, we introduce two new types of defenses against these attacks.

Functional and pharmacological role of the dopamine D4 receptor and its polymorphic variants

The functional and pharmacological significance of the dopamine D4 receptor (D4R) has remained the least well understood of all the dopamine receptor subtypes. Even more enigmatic has been the role of the very prevalent human DRD4 gene polymorphisms in the region that encodes the third intracellular loop of the receptor. The most common polymorphisms encode a D4R with 4 or 7 repeats of a proline-rich sequence of 16 amino acids (D4.4R and D4.7R). DRD4 polymorphisms have been associated with individual differences linked to impulse control-related neuropsychiatric disorders, with the most consistent associations established between the gene encoding D4.7R and attention-deficit hyperactivity disorder (ADHD) and substance use disorders. The function of D4R and its polymorphic variants is being revealed by addressing the role of receptor heteromerization and the relatively avidity of norepinephrine for D4R. We review the evidence conveying a significant and differential role of D4.4R and D4.7R in the dopaminergic and noradrenergic modulation of the frontal cortico-striatal pyramidal neuron, with implications for the moderation of constructs of impulsivity as personality traits. This differential role depends on their ability to confer different properties to adrenergic a2A receptor. (a2AR)-D4R heteromers and dopamine D2 receptor (D2R)-D4R heteromers, preferentially localized in the perisomatic region of the frontal cortical pyramidal neuron and its striatal terminals, respectively. We also review the evidence to support the D4R as a therapeutic target for ADHD and other impulse-control disorders, as well as for restless legs syndrome