111 research outputs found
Almost Optimal Pseudorandom Generators for Spherical Caps
Halfspaces or linear threshold functions are widely studied in complexity
theory, learning theory and algorithm design. In this work we study the natural
problem of constructing pseudorandom generators (PRGs) for halfspaces over the
sphere, aka spherical caps, which besides being interesting and basic geometric
objects, also arise frequently in the analysis of various randomized algorithms
(e.g., randomized rounding). We give an explicit PRG which fools spherical caps
within error and has an almost optimal seed-length of . For an inverse-polynomially
growing error , our generator has a seed-length optimal up to a
factor of . The most efficient PRG previously known (due
to Kane, 2012) requires a seed-length of in this
setting. We also obtain similar constructions to fool halfspaces with respect
to the Gaussian distribution.
Our construction and analysis are significantly different from previous works
on PRGs for halfspaces and build on the iterative dimension reduction ideas of
Kane et. al. (2011) and Celis et. al. (2013), the \emph{classical moment
problem} from probability theory and explicit constructions of \emph{orthogonal
designs} based on the seminal work of Bourgain and Gamburd (2011) on expansion
in Lie groups.Comment: 28 Pages (including the title page
Pseudorandomness via the discrete Fourier transform
We present a new approach to constructing unconditional pseudorandom
generators against classes of functions that involve computing a linear
function of the inputs. We give an explicit construction of a pseudorandom
generator that fools the discrete Fourier transforms of linear functions with
seed-length that is nearly logarithmic (up to polyloglog factors) in the input
size and the desired error parameter. Our result gives a single pseudorandom
generator that fools several important classes of tests computable in logspace
that have been considered in the literature, including halfspaces (over general
domains), modular tests and combinatorial shapes. For all these classes, our
generator is the first that achieves near logarithmic seed-length in both the
input length and the error parameter. Getting such a seed-length is a natural
challenge in its own right, which needs to be overcome in order to derandomize
RL - a central question in complexity theory.
Our construction combines ideas from a large body of prior work, ranging from
a classical construction of [NN93] to the recent gradually increasing
independence paradigm of [KMN11, CRSW13, GMRTV12], while also introducing some
novel analytic machinery which might find other applications
Quantified Derandomization of Linear Threshold Circuits
One of the prominent current challenges in complexity theory is the attempt
to prove lower bounds for , the class of constant-depth, polynomial-size
circuits with majority gates. Relying on the results of Williams (2013), an
appealing approach to prove such lower bounds is to construct a non-trivial
derandomization algorithm for . In this work we take a first step towards
the latter goal, by proving the first positive results regarding the
derandomization of circuits of depth .
Our first main result is a quantified derandomization algorithm for
circuits with a super-linear number of wires. Specifically, we construct an
algorithm that gets as input a circuit over input bits with
depth and wires, runs in almost-polynomial-time, and
distinguishes between the case that rejects at most inputs
and the case that accepts at most inputs. In fact, our
algorithm works even when the circuit is a linear threshold circuit, rather
than just a circuit (i.e., is a circuit with linear threshold gates,
which are stronger than majority gates).
Our second main result is that even a modest improvement of our quantified
derandomization algorithm would yield a non-trivial algorithm for standard
derandomization of all of , and would consequently imply that
. Specifically, if there exists a quantified
derandomization algorithm that gets as input a circuit with depth
and wires (rather than wires), runs in time at
most , and distinguishes between the case that rejects at
most inputs and the case that accepts at most
inputs, then there exists an algorithm with running time
for standard derandomization of .Comment: Changes in this revision: An additional result (a PRG for quantified
derandomization of depth-2 LTF circuits); rewrite of some of the exposition;
minor correction
Fast Optimal Locally Private Mean Estimation via Random Projections
We study the problem of locally private mean estimation of high-dimensional
vectors in the Euclidean ball. Existing algorithms for this problem either
incur sub-optimal error or have high communication and/or run-time complexity.
We propose a new algorithmic framework, ProjUnit, for private mean estimation
that yields algorithms that are computationally efficient, have low
communication complexity, and incur optimal error up to a -factor. Our
framework is deceptively simple: each randomizer projects its input to a random
low-dimensional subspace, normalizes the result, and then runs an optimal
algorithm such as PrivUnitG in the lower-dimensional space. In addition, we
show that, by appropriately correlating the random projection matrices across
devices, we can achieve fast server run-time. We mathematically analyze the
error of the algorithm in terms of properties of the random projections, and
study two instantiations. Lastly, our experiments for private mean estimation
and private federated learning demonstrate that our algorithms empirically
obtain nearly the same utility as optimal ones while having significantly lower
communication and computational cost.Comment: Added the correct github lin
Development of advanced geometric models and acceleration techniques for Monte Carlo simulation in Medical Physics
Els programes de simulació Monte Carlo de caràcter general s'utilitzen actualment en una gran varietat d'aplicacions.Tot i això, els models geomètrics implementats en la majoria de programes imposen certes limitacions a la forma dels objectes que es poden definir. Aquests models no són adequats per descriure les superfícies arbitràries que es troben en estructures anatòmiques o en certs aparells mèdics i, conseqüentment, algunes aplicacions que requereixen l'ús de models geomètrics molt detallats no poden ser acuradament estudiades amb aquests programes.L'objectiu d'aquesta tesi doctoral és el desenvolupament de models geomètrics i computacionals que facilitin la descripció dels objectes complexes que es troben en aplicacions de física mèdica. Concretament, dos nous programes de simulació Monte Carlo basats en PENELOPE han sigut desenvolupats. El primer programa, penEasy, utilitza un algoritme de caràcter general estructurat i inclou diversos models de fonts de radiació i detectors que permeten simular fàcilment un gran nombre d'aplicacions. Les noves rutines geomètriques utilitzades per aquest programa, penVox, extenen el model geomètric estàndard de PENELOPE, basat en superfícices quàdriques, per permetre la utilització d'objectes voxelitzats. Aquests objectes poden ser creats utilitzant la informació anatòmica obtinguda amb una tomografia computeritzada i, per tant, aquest model geomètric és útil per simular aplicacions que requereixen l'ús de l'anatomia de pacients reals (per exemple, la planificació radioterapèutica). El segon programa, penMesh, utilitza malles de triangles per definir la forma dels objectes simulats. Aquesta tècnica, que s'utilitza freqüentment en el camp del disseny per ordinador, permet representar superfícies arbitràries i és útil per simulacions que requereixen un gran detall en la descripció de la geometria, com per exemple l'obtenció d'imatges de raig x del cos humà.Per reduir els inconvenients causats pels llargs temps d'execució, els algoritmes implementats en els nous programes s'han accelerat utilitzant tècniques sofisticades, com per exemple una estructura octree. També s'ha desenvolupat un paquet de programari per a la paral·lelització de simulacions Monte Carlo, anomentat clonEasy, que redueix el temps real de càlcul de forma proporcional al nombre de processadors que s'utilitzen.Els programes de simulació que es presenten en aquesta tesi són gratuïts i de codi lliures. Aquests programes s'han provat en aplicacions realistes de física mèdica i s'han comparat amb altres programes i amb mesures experimentals.Per tant, actualment ja estan llestos per la seva distribució pública i per la seva aplicació a problemes reals.Monte Carlo simulation of radiation transport is currently applied in a large variety of areas. However, the geometric models implemented in most general-purpose codes impose limitations on the shape of the objects that can be defined. These models are not well suited to represent the free-form (i.e., arbitrary) shapes found in anatomic structures or complex medical devices. As a result, some clinical applications that require the use of highly detailed phantoms can not be properly addressed.This thesis is devoted to the development of advanced geometric models and accelration techniques that facilitate the use of state-of-the-art Monte Carlo simulation in medical physics applications involving detailed anatomical phantoms. To this end, two new codes, based on the PENELOPE package, have been developed. The first code, penEasy, implements a modular, general-purpose main program and provides various source models and tallies that can be readily used to simulate a wide spectrum of problems. Its associated geometry routines, penVox, extend the standard PENELOPE geometry, based on quadric surfaces, to allow the definition of voxelised phantoms. This kind of phantoms can be generated using the information provided by a computed tomography and, therefore, penVox is convenient for simulating problems that require the use of the anatomy of real patients (e.g., radiotherapy treatment planning). The second code, penMesh, utilises closed triangle meshes to define the boundary of each simulated object. This approach, which is frequently used in computer graphics and computer-aided design, makes it possible to represent arbitrary surfaces and it is suitable for simulations requiring a high anatomical detail (e.g., medical imaging).A set of software tools for the parallelisation of Monte Carlo simulations, clonEasy, has also been developed. These tools can reduce the simulation time by a factor that is roughly proportional to the number of processors available and, therefore, facilitate the study of complex settings that may require unaffordable execution times in a sequential simulation.The computer codes presented in this thesis have been tested in realistic medical physics applications and compared with other Monte Carlo codes and experimental data. Therefore, these codes are ready to be publicly distributed as free and open software and applied to real-life problems.Postprint (published version
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