Balanced Group Convolution: An Improved Group Convolution Based on Approximability Estimates

The performance of neural networks has been significantly improved by increasing the number of channels in convolutional layers. However, this increase in performance comes with a higher computational cost, resulting in numerous studies focused on reducing it. One promising approach to address this issue is group convolution, which effectively reduces the computational cost by grouping channels. However, to the best of our knowledge, there has been no theoretical analysis on how well the group convolution approximates the standard convolution. In this paper, we mathematically analyze the approximation of the group convolution to the standard convolution with respect to the number of groups. Furthermore, we propose a novel variant of the group convolution called balanced group convolution, which shows a higher approximation with a small additional computational cost. We provide experimental results that validate our theoretical findings and demonstrate the superior performance of the balanced group convolution over other variants of group convolution.Comment: 26pages, 2 figure

SAR by MS for Functional Genomics (Structure-Activity Relation by Mass Spectrometry)

Large-scale functional genomics will require fast, high-throughput experimental techniques, coupled with sophisticated computer algorithms for data analysis and experiment planning. In this paper, we introduce a combined experimental-computational protocol called Structure-Activity Relation by Mass Spectrometry (SAR by MS), which can be used to elucidate the function of protein-DNA or protein-protein complexes. We present algorithms for SAR by MS and analyze their complexity. Carefully-designed Matrix-Assisted Laser Desorption/Ionization Time-Of-Flight (MALDI TOF) and Electrospray Ionization (ESI) assays require only femtomolar samples, take only microseconds per spectrum to record, enjoy a resolution of up to one dalton in $10^6$, and (in the case of MALDI) can operate on protein complexes up to a megadalton in mass. Hence, the technique is attractive for high-throughput functional genomics. In SAR by MS, selected residues or nucleosides are 2H-, 13C-, and/or 15N-labeled. Second, the complex is crosslinked. Third, the complex is cleaved with proteases and/or endonucleases. Depending on the binding mode, some cleavage sites will be shielded by the crosslinking. Finally, a mass spectrum of the resulting fragments is obtained and analyzed. The last step is the Data Analysis phase, in which the mass signatures are interpreted to obtain constraints on the functional binding mode. Experiment Planning entails deciding what labeling strategy and cleaving agents to employ, so as to minimize mass degeneracy and spectral overlap, in order that the constraints derived in data analysis yield a small number of binding hypotheses. A number of combinatorial and algorithmic questions arise in deriving algorithms for both Experiment Planning and Data Analysis. We explore the complexity of these problems, obtaining upper and lower bounds. Experimental results are reported from an implementation of our algorithms

On the Stability of Distribution Topologies in Peer-to-Peer Live Streaming Systems

Peer-to-Peer Live-Streaming-Systeme sind ständigen Störungen ausgesetzt.Insbesondere ermöglichen unzuverlässige Teilnehmer Ausfälle und Angriffe, welche überraschend Peers aus dem System entfernen. Die Folgen solcher Vorfälle werden großteils von der Verteilungstopologie bestimmt, d.h. der Kommunikationsstruktur zwischen den Peers.In dieser Arbeit analysieren wir Optimierungsprobleme welche bei der Betrachtung von Stabilitätsbegriffen für solche Verteilungstopologien auftreten. Dabei werden sowohl Angriffe als auch unkoordinierte Ausfälle berücksichtigt.Zunächst untersuchen wir die Berechnungskomplexität und Approximierbarkeit des Problems resourcen-effiziente Angriffe zu bestimmen. Dies demonstriert Beschränkungen in den Planungsmöglichkeiten von Angreifern und zeigt inwieweit die Topologieparameter die Schwierigkeit solcher Angriffsrobleme beeinflussen. Anschließend studieren wir Topologieformationsprobleme. Dabei sind Topologieparameter vorgegeben und es muss eine passende Verteilungstopologie gefunden werden. Ziel ist es Topologien zu erzeugen, welche den durch Angriffe mit beliebigen Parametern erzeugbaren maximalen Schaden minimieren.Wir identifizieren notwendige und hinreichende Eigenschaften solcher Verteilungstopologien. Dies führt zu mathematisch fundierten Zielstellungen für das Topologie-Management von Peer-to-Peer Live-Streaming-Systemen.Wir zeigen zwei große Klassen effizient konstruierbarer Verteilungstopologien, welche den maximal möglichen, durch Angriffe verursachten Paketverlust minimieren. Zusätzlich beweisen wir, dass die Bestimmung dieser Eigenschaft für beliebige Topologien coNP-vollständig ist.Soll die maximale Anzahl von Peers minimiert werden, bei denen ein Angriff zu ungenügender Stream-Qualität führt, ändern sich die Anforderungen an Verteilungstopologien. Wir zeigen, dass dieses Topologieformationsproblem eng mit offenen Problemen aus Design- und Kodierungstheorie verwandt ist.Schließlich analysieren wir Verteilungstopologien die den durch unkoordinierte Ausfälle zu erwartetenden Paketverlust minimieren. Wir zeigen Eigenschaften und Existenzbedingungen. Außerdem bestimmen wir die Berechnungskomplexität des Auffindens solcher Topologien. Unsere Ergebnisse liefern Richtlinien für das Topologie-Management von Peer-to-Peer Live-Streaming-Systemen und zeigen auf, welche Stabilitätsziele effizient erreicht werden können.The stability of peer-to-peer live streaming systems is constantly challenged. Especially, the unreliability and vulnerability of their participants allows for failures and attacks suddenly disabling certain sets of peers. The consequences of such events are largely determined by the distribution topology, i.e., the pattern of communication between the peers.In this thesis, we analyze a broad range of optimization problems concerning the stability of distribution topologies. For this, we discuss notions of stability against both attacks and failures.At first, we investigate the computational complexity and approximability of finding resource-efficient attacks. This allows to point out limitations of an attacker's planning capabilities and demonstrates the influence of the chosen system parameters on the hardness of such attack problems.Then, we turn to study topology formation problems. Here, a set of topology parameters is given and the task consists in finding an eligible distribution topology. In particular, it has to minimize the maximum damage achievable by attacks with arbitrary attack parameters.We identify necessary and sufficient conditions on attack-stable distribution topologies. Thereby, we give mathematically sound guidelines for the topology management of peer-to-peer live streaming systems.We find large classes of efficiently-constructable topologies minimizing the system-wide packet loss under attacks. Additionally, we show that determining this feature for arbitrary topologies is coNP-complete.Considering topologies minimizing the maximum number of peers for which an attack leads to a heavy decrease in perceived streaming quality, the requirements change. Here, we show that the corresponding topology formation problem is closely related to long-standing open problems of Design and Coding Theory.Finally, we study topologies minimizing the expected packet loss due to uncoordinated peer failures. We investigate properties and existence conditions of such topologies. Furthermore, we determine the computational complexity of constructing them.Our results provide guidelines for the topology management of peer-to-peer live streaming systems and mathematically determine which goals can be achieved efficiently

Multiwinner Elections with Diversity Constraints

We develop a model of multiwinner elections that combines performance-based measures of the quality of the committee (such as, e.g., Borda scores of the committee members) with diversity constraints. Specifically, we assume that the candidates have certain attributes (such as being a male or a female, being junior or senior, etc.) and the goal is to elect a committee that, on the one hand, has as high a score regarding a given performance measure, but that, on the other hand, meets certain requirements (e.g., of the form "at least $30\%$ of the committee members are junior candidates and at least $40\%$ are females"). We analyze the computational complexity of computing winning committees in this model, obtaining polynomial-time algorithms (exact and approximate) and NP-hardness results. We focus on several natural classes of voting rules and diversity constraints.Comment: A short version of this paper appears in the proceedings of AAAI-1