scholarly journals Sign-rank Can Increase under Intersection

2021 ◽  
Vol 13 (4) ◽  
pp. 1-17
Author(s):  
Mark Bun ◽  
Nikhil S. Mande ◽  
Justin Thaler
Keyword(s):  
Communication Complexity ◽  
Function Class ◽  
Time Algorithm ◽  
Constant Factor ◽  
Pac Learning ◽  
New Techniques ◽  
Know How ◽  
Communication Problem ◽  
Analytic Complexity ◽  
Class Of Functions

The communication class UPP cc is a communication analog of the Turing Machine complexity class PP . It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds. For a communication problem f , let f ∧ f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP cc ( f ) = O (log n ), and UPP cc ( f ∧ f ) = Θ (log 2 n ). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPP cc , the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity n Omega Ω(log n ) . This matches an upper bound of (Klivans, O’Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time.

scholarly journals Learning Complexity vs Communication Complexity

2009 ◽  
Vol 18 (1-2) ◽  
pp. 227-245 ◽  
Author(s):  
NATI LINIAL ◽  
ADI SHRAIBMAN
Keyword(s):  
Communication Complexity ◽  
Machine Learning Algorithms ◽  
Support Vector ◽  
Polynomial Hierarchy ◽  
Open Problems ◽  
Multiplicative Constant ◽  
Large Margin ◽  
Vector Machines ◽  
Large Margin Classifiers ◽  
Class Of Functions

This paper has two main focal points. We first consider an important class of machine learning algorithms: large margin classifiers, such as Support Vector Machines. The notion of margin complexity quantifies the extent to which a given class of functions can be learned by large margin classifiers. We prove that up to a small multiplicative constant, margin complexity is equal to the inverse of discrepancy. This establishes a strong tie between seemingly very different notions from two distinct areas.In the same way that matrix rigidity is related to rank, we introduce the notion of rigidity of margin complexity. We prove that sign matrices with small margin complexity rigidity are very rare. This leads to the question of proving lower bounds on the rigidity of margin complexity. Quite surprisingly, this question turns out to be closely related to basic open problems in communication complexity, e.g., whether PSPACE can be separated from the polynomial hierarchy in communication complexity.Communication is a key ingredient in many types of learning. This explains the relations between the field of learning theory and that of communication complexity [6, l0, 16, 26]. The results of this paper constitute another link in this rich web of relations. These new results have already been applied toward the solution of several open problems in communication complexity [18, 20, 29].

scholarly journals Maximin-Aware Allocations of Indivisible Goods

2019 ◽  
Author(s):  
Hau Chan ◽  
Jing Chen ◽  
Bo Li ◽  
Xiaowei Wu
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Indivisible Goods ◽  
Know How

We study envy-free allocations of indivisible goods to agents in settings where each agent is unaware of the goods allocated to other agents. In particular, we propose the maximin aware (MMA) fairness measure, which guarantees that every agent, given the bundle allocated to her, is aware that she does not envy at least one other agent, even if she does not know how the other goods are distributed among other agents. We also introduce two of its relaxations, and discuss their egalitarian guarantee and existence. Finally, we present a polynomial-time algorithm, which computes an allocation that approximately satisfies MMA or its relaxations. Interestingly, the returned allocation is also 1/2-approximate EFX when all agents have sub- additive valuations, which improves the algorithm in [Plaut and Roughgarden, 2018].

scholarly journals Ultrasound guided biopsy, a gold standard diagnostical test of the prostate cancer

2005 ◽  
Vol 52 (4) ◽  
pp. 23-26 ◽  
Author(s):  
I. Romics
Keyword(s):  
Prostate Cancer ◽  
Gold Standard ◽  
Middle Part ◽  
Time Span ◽  
Targeted Biopsy ◽  
Ultrasound Guided ◽  
New Techniques ◽  
Know How ◽  
Preoperative Antibiotics ◽  
Palpable Nodule

The author discusses preparations for ultrasound guided prostate biopsy, its technique conditions and the process of performing a biopsy. Every author proposes the use of preoperative antibiotics based prophylaxis. Differences may be found in the type, dosage and the time span of preoperative application. For anesthesia mostly lidocaine was proposed, which may be a gel applied in the rectum or used in the form a prostate infiltrate. The widest debate goes on in respect of defining the number of biopsies needed. Recently 8 or rather 10 samples are proposed to be taken. Twelve biopsies do offer an advantage compared to 6 although in case of 8 this isn?t so. According to the site of sample taking the apex, the base and the middle part are proposed. In case of a palpable nodule or any lesion, made visible by TRUS an additional, targeted, biopsy has to be performed. Certain new techniques like the 3D Doppler, contrast, intermittent and others shall also be presented. A repeated biopsy shall be necessary in case of PIN atypia, beyond that the author also discusses other indications for a repeated biopsy. We may expect the occurrence of direct postoperative complications and it is necessary to know how to treat these.

scholarly journals Subclasses of Bi-Univalent Functions Defined by Frasin Differential Operator

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 783 ◽  
Author(s):  
Ibtisam Aldawish ◽  
Tariq Al-Hawary ◽  
B. A. Frasin
Keyword(s):  
Analytic Function ◽  
Differential Operator ◽  
Unit Disc ◽  
Univalent Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
Open Unit Disc ◽  
Class A ◽  
Class Of Functions

Let Ω denote the class of functions f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ belonging to the normalized analytic function class A in the open unit disk U = z : z < 1 , which are bi-univalent in U , that is, both the function f and its inverse f − 1 are univalent in U . In this paper, we introduce and investigate two new subclasses of the function class Ω of bi-univalent functions defined in the open unit disc U , which are associated with a new differential operator of analytic functions involving binomial series. Furthermore, we find estimates on the Taylor–Maclaurin coefficients | a 2 | and | a 3 | for functions in these new subclasses. Several (known or new) consequences of the results are also pointed out.

Informational Robustness in Intertemporal Pricing

2020 ◽  
Author(s):  
Jonathan Libgober ◽  
Xiaosheng Mu
Keyword(s):  
Arrival Process ◽  
Durable Good ◽  
Worst Case ◽  
New Techniques ◽  
Know How ◽  
Dynamic Monopoly ◽  
Intertemporal Pricing ◽  
The Face ◽  
Arrival Processes ◽  
Constant Prices

Abstract We introduce a robust approach to study dynamic monopoly pricing of a durable good in the face of buyer learning. A buyer receives information about her willingness-to-pay for the seller’s product over time, and decides when to make a one-time purchase. The seller does not know how the buyer learns but commits to a pricing strategy to maximize profits against the worst-case information arrival process. We show that a constant price path delivers the robustly optimal profit, with profit and price both lower than under known values. Thus, under the robust objective, intertemporal incentives do not arise at the optimum, despite the possibility for information arrival to influence the timing of purchases. We delineate whether constant prices remain optimal (or not) when the seller seeks robustness against a subset of information arrival processes. As part of the analysis, we develop new techniques to study dynamic Bayesian persuasion.

scholarly journals Computing the Partition Function for Perfect Matchings in a Hypergraph

2011 ◽  
Vol 20 (6) ◽  
pp. 815-835 ◽  
Author(s):  
ALEXANDER BARVINOK ◽  
ALEX SAMORODNITSKY
Keyword(s):  
Partition Function ◽  
Polynomial Time ◽  
Pairwise Disjoint ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Perfect Matchings ◽  
Constant Factor ◽  
Higher Dimensional ◽  
Element Set

Given non-negative weightswSon thek-subsetsSof akm-element setV, we consider the sum of the productswS1⋅⋅⋅wSmover all partitionsV=S1∪ ⋅⋅⋅ ∪Sminto pairwise disjointk-subsetsSi. When the weightswSare positive and within a constant factor of each other, fixed in advance, we present a simple polynomial-time algorithm to approximate the sum within a polynomial inmfactor. In the process, we obtain higher-dimensional versions of the van der Waerden and Bregman–Minc bounds for permanents. We also discuss applications to counting of perfect and nearly perfect matchings in hypergraphs.

EMBEDDING THE DOUBLE CIRCLE IN A SQUARE GRID OF MINIMUM SIZE

2014 ◽  
Vol 24 (03) ◽  
pp. 247-258 ◽  
Author(s):  
S. BEREG ◽  
R. FABILA-MONROY ◽  
D. FLORES-PEÑALOZA ◽  
M. A. LOPEZ ◽  
P. PÉREZ-LANTERO
Keyword(s):  
Time Algorithm ◽  
Constant Factor ◽  
Grid Size ◽  
Lattice Points ◽  
Minimum Size ◽  
Small Integer ◽  
Square Grid ◽  
Similar Construction ◽  
Asymptotic Size ◽  
Point Set

In 1926, Jarník investigated the drawing of a curve that visits a large number of lattice points relative to its curvature. To this end, he constructed a convex n-gon with vertices on a “small” integer grid [0, c.n3/2]2, where c > 0 is a constant, and proved that this grid size is optimal up to a constant factor. We consider a similar construction for the double circle of 2n points and prove that it can be embedded in a grid of the same asymptotic size. Moreover, we give an O(n)-time algorithm to generate the corresponding point set.

scholarly journals The hardest halfspace

2021 ◽  
Vol 30 (2) ◽  
Author(s):  
Alexander A. Sherstov
Keyword(s):  
Lower Bounds ◽  
Rational Approximation ◽  
Communication Complexity ◽  
Rational Functions ◽  
Explicit Construction ◽  
Upper Bounds ◽  
Infinity Norm ◽  
Communication Problem

AbstractWe study the approximation of halfspaces $$h:\{0,1\}^n\to\{0,1\}$$ h : { 0 , 1 } n → { 0 , 1 } in the infinity norm by polynomials and rational functions of any given degree. Our main result is an explicit construction of the “hardest” halfspace, for which we prove polynomial and rational approximation lower bounds that match the trivial upper bounds achievable for all halfspaces. This completes a lengthy line of work started by Myhill and Kautz (1961). As an application, we construct a communication problem that achieves essentially the largest possible separation, of O(n) versus $$2^{-\Omega(n)}$$ 2 - Ω ( n ) , between the sign-rank and discrepancy. Equivalently, our problem exhibits a gap of log n versus $$\Omega(n)$$ Ω ( n ) between the communication complexity with unbounded versus weakly unbounded error, improving quadratically on previous constructions and completing a line of work started by Babai, Frankl, and Simon (FOCS 1986). Our results further generalize to the k-party number-on-the-forehead model, where we obtain an explicit separation of log n versus $$\Omega(n/4^{n})$$ Ω ( n / 4 n ) for communication with unbounded versus weakly unbounded error.

Relatie Tussen Doelstellingen-formulering en Didaktische Werkvormen.

1977 ◽  
Vol 3 ◽  
pp. 114-132
Author(s):  
W.P.B.M. Welsing
Keyword(s):  
Learning Process ◽  
Threshold Level ◽  
Functional Approach ◽  
Teaching Material ◽  
Teaching Techniques ◽  
New Techniques ◽  
Know How ◽  
Definition Of ◽  
The Individual ◽  
The Way

In the introduction the author makes three points he considers import-ant, although they have no direct bearing on the main topic. 1. Much of the criticism on The Threshold Level in the Netherlands unfortunately focuses on the choice of functions and the list of structures and vocabulary. The New direction The Threshold Level suggests is not always fully appreciated. 2. Teachers that know how to motivate their pupils, get results what-ever techniques they use. 3. All those who are actively engaged in the definition of objectives should at all times be aware of the consequences for the teachers who will have to implement them. If we accept functional notional objectives that take the needs of the individual learner into account, we shall have to realize that it will be impossible to give exact definitions of any level. The word "approach" is therefore very adequate. Functional objectives show the way to go and give therefore more support to the teachers in the planning of their lessons. The approach will also stimulate motivation. For testers, however, the functional approach will cause more problems. Functional objectives are defined in terms of behaviour. This also determines the nature of the teaching techniques to be applied. Skating can only be learned by practice! Theoretical information about the language may help the learner, but only in so far as it is relevant for the learning process. The same is true for the way in which this information is given. Teaching material used to be graded carefully with the structional approach. With the functional approach the ordering will have to be done on a different basis. New techniques will have to be developed. The author concludes by giving examples and showing that in F.L.T., whatever the nature of the objectives, the teaching techniques should always be communicative.

COMPUTING THE DISCRETE FRÉCHET DISTANCE WITH IMPRECISE INPUT

2012 ◽  
Vol 22 (01) ◽  
pp. 27-44 ◽  
Author(s):  
HEE-KAP AHN ◽  
CHRISTIAN KNAUER ◽  
MARC SCHERFENBERG ◽  
LENA SCHLIPF ◽  
ANTOINE VIGNERON
Keyword(s):  
Time Algorithm ◽  
Constant Factor ◽  
Fréchet Distance ◽  
Planar Case ◽  
Running Time ◽  
Frechet Distance ◽  
Axis Parallel ◽  
Polygonal Curves ◽  
Orthogonal Case ◽  
Improved Algorithm

We consider the problem of computing the discrete Fréchet distance between two polygonal curves when their vertices are imprecise. An imprecise point is given by a region and this point could lie anywhere within this region. By modelling imprecise points as balls in dimension d, we present an algorithm for this problem that returns in time 2O(d2) m2n2 log 2(mn) the minimum Fréchet distance between two imprecise polygonal curves with n and m vertices, respectively. We give an improved algorithm for the planar case with running time O(mn log 3(mn) + (m2+n2) log (mn)). In the d-dimensional orthogonal case, where points are modelled as axis-parallel boxes, and we use the L∞ distance, we give an O(dmn log (dmn))-time algorithm. We also give efficient O(dmn)-time algorithms to approximate the maximum Fréchet distance, as well as the minimum and maximum Fréchet distance under translation. These algorithms achieve constant factor approximation ratios in "realistic" settings (such as when the radii of the balls modelling the imprecise points are roughly of the same size).

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