One of the drawbacks of A-DPS compared to learned fixed sampling schemes is it higher amount of c... more One of the drawbacks of A-DPS compared to learned fixed sampling schemes is it higher amount of computational complexity. The main source of this complexity is the unrolling of iterations, leading to a computational complexity of O(I) = O(M/ρ). Although we set ρ equal to 1 in all our experiments, one can in fact seamlessly interpolate between A-DPS and DPS by choosing 1 ≤ ρ ≤ M. This constitutes a trade-off between computational complexity and adaptation rate. We leave further exploration of this trade-off to future work. We can also express computational complexity in terms of run-time on a machine, in our case a GeForce GTX 1080 Ti. A comparison of DPS and A-DPS in terms of training time per epoch can be seen in Fig. 8. We can see that the training time for A-DPS increases for higher sampling ratios where it needs to unroll through more iterations. By combining the results from Fig. 2 (in the main body of the paper) and Fig. 8, one can make a trade-off between run-time and accuracy. Where A-DPS achieves higher accuracy for stricter sampling regimes, while at the same time not increasing run-time by a lot.
Proceedings Fifth Annual Structure in Complexity Theory Conference
Two aspects of the structure of complete sets for NE and larger nondeterministic time classes are... more Two aspects of the structure of complete sets for NE and larger nondeterministic time classes are surveyed. First, differences between complete sets arising from various polynomial-time reductions are proved. Immunity properties of these complete sets are then considered. It is shown that NE-complete sets (and their complements) have dense E subsets and dense UP subsets. All of these results hold
It is shown that determining whether a quantum computation has a non-zero probability of acceptin... more It is shown that determining whether a quantum computation has a non-zero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness result also applies to determining in general whether a given quantum basis state appears with nonzero amplitude in a superposition, or whether a given quantum bit has positive expectation value at the end of a quantum computation. 1
Fix any λ ∈ C. We say that a set S ⊆ C is λ-convex if, whenever a and b are in S, the point (1 − ... more Fix any λ ∈ C. We say that a set S ⊆ C is λ-convex if, whenever a and b are in S, the point (1 − λ)a + λb is also in S. If S is also (topologically) closed, then we say that S is λ-clonvex. We investigate the properties of λ-convex and λ-clonvex sets and prove a number of facts about them. Letting R λ ⊆ C be the least λ-clonvex superset of {0, 1}, we show that if R λ is convex in the usual sense, then R λ must be either [0, 1] or R or C, depending on λ. We investigate which λ make R λ convex, derive a number of conditions equivalent to R λ being convex, and give several conditions sufficient for R λ to be convex or not convex; in particular, we show that R λ is either convex or uniformly discrete. Letting C := {λ ∈ C | R λ is convex}, we show that C is open and contains all transcendental λ. It follows that C \ C is countable, closed, and nowhere dense. We also give a sufficient condition on λ for R λ and some other related λ-convex sets to be discrete by introducing the notion of a strong PV number. These conditions give rise to a number of periodic and aperiodic Meyer sets (the latter sometimes known as "quasicrystals"). The paper is in four parts. Part I describes basic properties of λ-convex and λ-clonvex sets, including convexity versus uniform discreteness. Part II explores the connections between λconvex sets and quasicrystals and displays a number of such sets, including several with dihedral symmetry. Part III generalizes a result from Part I, and Part IV contains our conclusions and open problems. Our work combines elementary concepts and techniques from algebra and plane geometry.
In this paper we consider splitting theorems for 18-recursion theory where/3 is a weakly admi~ibl... more In this paper we consider splitting theorems for 18-recursion theory where/3 is a weakly admi~ible ordinal. Two theorems are p~oved in this setting. The first is a splitting theorem for simple subsets of /3", the second a splitting theorem for certain regular ~-r,e. sets, The splitting theorem for a-recursion theory, where a is a Zt-admissible ordinal, was proved by Shore [14] where he showed how to split a regular a-r.e, set A into two a-r.e, sets B and C whose join has the same ~x-degree as A, while the s-degrees of B and C are both less than the a-degree of A. Essential to Shore's proof is the method of blocking in which a block of requirements is considered as a single requirement ,~n a priority construction, One immediate obstacle to adapting Shore's proof for a-recursion theory to fl-recursion theory is the lack of a regular sets theorem. Shore's proof enabled him to split all regular a-r,e., non-a-recursive sets. The regular sets theorem of Sacks [12] then enabled Shore to cJnclude that he could split sets in any non-a-recursive a-r,e, degree. Wolfgang Maass [10] has shown that such a regular sets theorem does not hold for all weakly admissible ordinals. Thus, in trying ~o generalize the splitting theorem to this setting, one is led to deal with sets which are not necessarily regular. Two alternate courses for proving a splitting theorem for/3-recursion theory are suggested by the above difficulty. The first is to meet the problem head-on and to try to prove a splitting theorem for non-regular sets. This is done in Section 2 below where, with the additional a~umption that /3 * is a regular /3-cardinal, simple tamely-fl-r.e, subsets of /3" are split, The proof is quite different from Shore's and is more in the spirit of Owings' splitting theorem [11] for metarecursion theory. A second way of approaching this problem is to assume that the B-r.e. set we are trying to split is regular and to later examine which fl-r.e, degrees contain such sets, Another problem arises here in trying to generalize the proof from * The results of this paper arc from the author's Ph.D~ thesis written at M.I.T. in 1978. I would like to thank my th~is advisor. Professor Gerald Sacks. for his encouragement and guidance during the preparation of this wx~rk.
Proceedings 35th Annual Symposium on Foundations of Computer Science
We present a polynomial time online learning algorithm that learns any discretized geometric conc... more We present a polynomial time online learning algorithm that learns any discretized geometric concept generated from any number of halfspaces with any number of known (to the learner) slopes in a constant dimensional space. In particular, our algorithm learns (from equivalence queries only) unions of discretized axis-parallel rectangles in a constant dimensional space in polynomial time. The algorithm also runs in polynomial time an 1 if the teacher lies on 1 counterexamples. We then show a PAC-learning algorithm for the above discretized geometric concept when the example oracle lies on the labels of the examples with a fixed probability p 5 3that runs in polynomial time also with r. We use these methods, as well as a bounded version of the finite injury priority method, to construct algorithms f o r learning several classes of rectangles. In particular we design eficient algorithms for learning several classes of unions of discretized axis-parallel rectangles in either arbitrary dimensional spaces or constant dimensional spaces.
We define and construct efficient depth-universal and almostsize-universal quantum circuits. Such... more We define and construct efficient depth-universal and almostsize-universal quantum circuits. Such circuits can be viewed as generalpurpose simulators for central quantum circuit classes and used to capture the computational power of the simulated class. For depth we construct universal circuits whose depth is the same order as the circuits being simulated. For size, there is a log factor blow-up in the universal circuits constructed here which is nearly optimal for polynomial size circuits. {(dbera|homer)}@cs.bu.edu.
Abstract. We study the problem of finding a hidden code k in the domain (1,..., my* in the presen... more Abstract. We study the problem of finding a hidden code k in the domain (1,..., my* in the presence of an oracle which, for any z in the domain, answers a pair of numbers a(z, k) and b(z, k) such that a(z, k) is the number of components coinciding in z and k and, b(z, k) is the sum of ...
We demonstrate differences between reducibilities and corresponding completeness notions for nond... more We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomial-time bounded (even logarithmic-space bounded) reducibilities turn out to be different for any class containing NE. For space classes the completeness notions under logspace reducibilities can be separated for any class properly containing LOGSPACE. Key observation in obtaining the separations is the honesty property of reductions, which was recently observed to hold for the time classes and can be shown to hold for space classes.
We consider under the assumption PINP questions concerning the structure of the lattice of NP set... more We consider under the assumption PINP questions concerning the structure of the lattice of NP sets together with the sublattice P. We show that two questions which are slightly more complex than the known splitting properties of this lattice cannot be settled by arguments which relativize. The two questions which we consider are whether every infinite NP set contains an infinite P subset and whether there exists an NP-simple set. We construct several oracles, all of which make P I NP, and which in addition make the above-mentioned statements either true or false. In particular we give a positive answer to the question, raised by Bennett and Gill (1981), whether an oracle B exists making PBINPt and such that every infinite set in NPB has an infinite subset in PB. The constructions of the oracles are finite injury priority arguments.
The existence of minimal degrees is investigated for several polynomial reducibilities. It is sho... more The existence of minimal degrees is investigated for several polynomial reducibilities. It is shown that no set has minimal degree with respect to polynomial many-one or Turing reducibility. This extends a result of Ladner in which only recursive sets are considered. A polynomial reducibility ≤<supscrpt><italic>h</italic></supscrpt><subscrpt><italic>T</italic></subscrpt> is defined. This reducibility is a strengthening of polynomial Turing reducibility, and its properties relate to the P = ? NP question. For this new reducibility, a set of minimal degree is constructed under the assumption that P = NP. However, the set constructed is nonrecursive, and it is shown that no recursive set is of minimal ≤ <supscrpt><italic>h</italic></supscrpt><subscrpt><italic>T</italic></subscrpt> degree.
Journal of Parallel and Distributed Computing, 1997
We develop and experiment with a new parallel algorithm to approximate the maximum weight cut in ... more We develop and experiment with a new parallel algorithm to approximate the maximum weight cut in a weighted undirected graph. Our implementation starts with the recent (serial) algorithm of Goemans and Williamson for this problem. We consider several different versions of this algorithm, varying the interior-point part of the algorithm in order to optimize the parallel efficiency of our method. Our work aims for an efficient, practical formulation of the algorithm with closeto-optimal parallelization. We analyze our parallel algorithm in the LogP model and predict linear speedup for a wide range of the parameters. We have implemented the algorithm using the message passing interface (MPI) and run it on several parallel machines. In particular, we present performance measurements on the IBM SP2, the Connection Machine CM5, and a cluster of workstations. We observe that the measured speedups are predicted well by our analysis in the LogP model. Finally, we test our implementation on several large graphs (up to 13,000 vertices), particularly on large instances of the Ising model.
One of the drawbacks of A-DPS compared to learned fixed sampling schemes is it higher amount of c... more One of the drawbacks of A-DPS compared to learned fixed sampling schemes is it higher amount of computational complexity. The main source of this complexity is the unrolling of iterations, leading to a computational complexity of O(I) = O(M/ρ). Although we set ρ equal to 1 in all our experiments, one can in fact seamlessly interpolate between A-DPS and DPS by choosing 1 ≤ ρ ≤ M. This constitutes a trade-off between computational complexity and adaptation rate. We leave further exploration of this trade-off to future work. We can also express computational complexity in terms of run-time on a machine, in our case a GeForce GTX 1080 Ti. A comparison of DPS and A-DPS in terms of training time per epoch can be seen in Fig. 8. We can see that the training time for A-DPS increases for higher sampling ratios where it needs to unroll through more iterations. By combining the results from Fig. 2 (in the main body of the paper) and Fig. 8, one can make a trade-off between run-time and accuracy. Where A-DPS achieves higher accuracy for stricter sampling regimes, while at the same time not increasing run-time by a lot.
Proceedings Fifth Annual Structure in Complexity Theory Conference
Two aspects of the structure of complete sets for NE and larger nondeterministic time classes are... more Two aspects of the structure of complete sets for NE and larger nondeterministic time classes are surveyed. First, differences between complete sets arising from various polynomial-time reductions are proved. Immunity properties of these complete sets are then considered. It is shown that NE-complete sets (and their complements) have dense E subsets and dense UP subsets. All of these results hold
It is shown that determining whether a quantum computation has a non-zero probability of acceptin... more It is shown that determining whether a quantum computation has a non-zero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness result also applies to determining in general whether a given quantum basis state appears with nonzero amplitude in a superposition, or whether a given quantum bit has positive expectation value at the end of a quantum computation. 1
Fix any λ ∈ C. We say that a set S ⊆ C is λ-convex if, whenever a and b are in S, the point (1 − ... more Fix any λ ∈ C. We say that a set S ⊆ C is λ-convex if, whenever a and b are in S, the point (1 − λ)a + λb is also in S. If S is also (topologically) closed, then we say that S is λ-clonvex. We investigate the properties of λ-convex and λ-clonvex sets and prove a number of facts about them. Letting R λ ⊆ C be the least λ-clonvex superset of {0, 1}, we show that if R λ is convex in the usual sense, then R λ must be either [0, 1] or R or C, depending on λ. We investigate which λ make R λ convex, derive a number of conditions equivalent to R λ being convex, and give several conditions sufficient for R λ to be convex or not convex; in particular, we show that R λ is either convex or uniformly discrete. Letting C := {λ ∈ C | R λ is convex}, we show that C is open and contains all transcendental λ. It follows that C \ C is countable, closed, and nowhere dense. We also give a sufficient condition on λ for R λ and some other related λ-convex sets to be discrete by introducing the notion of a strong PV number. These conditions give rise to a number of periodic and aperiodic Meyer sets (the latter sometimes known as "quasicrystals"). The paper is in four parts. Part I describes basic properties of λ-convex and λ-clonvex sets, including convexity versus uniform discreteness. Part II explores the connections between λconvex sets and quasicrystals and displays a number of such sets, including several with dihedral symmetry. Part III generalizes a result from Part I, and Part IV contains our conclusions and open problems. Our work combines elementary concepts and techniques from algebra and plane geometry.
In this paper we consider splitting theorems for 18-recursion theory where/3 is a weakly admi~ibl... more In this paper we consider splitting theorems for 18-recursion theory where/3 is a weakly admi~ible ordinal. Two theorems are p~oved in this setting. The first is a splitting theorem for simple subsets of /3", the second a splitting theorem for certain regular ~-r,e. sets, The splitting theorem for a-recursion theory, where a is a Zt-admissible ordinal, was proved by Shore [14] where he showed how to split a regular a-r.e, set A into two a-r.e, sets B and C whose join has the same ~x-degree as A, while the s-degrees of B and C are both less than the a-degree of A. Essential to Shore's proof is the method of blocking in which a block of requirements is considered as a single requirement ,~n a priority construction, One immediate obstacle to adapting Shore's proof for a-recursion theory to fl-recursion theory is the lack of a regular sets theorem. Shore's proof enabled him to split all regular a-r,e., non-a-recursive sets. The regular sets theorem of Sacks [12] then enabled Shore to cJnclude that he could split sets in any non-a-recursive a-r,e, degree. Wolfgang Maass [10] has shown that such a regular sets theorem does not hold for all weakly admissible ordinals. Thus, in trying ~o generalize the splitting theorem to this setting, one is led to deal with sets which are not necessarily regular. Two alternate courses for proving a splitting theorem for/3-recursion theory are suggested by the above difficulty. The first is to meet the problem head-on and to try to prove a splitting theorem for non-regular sets. This is done in Section 2 below where, with the additional a~umption that /3 * is a regular /3-cardinal, simple tamely-fl-r.e, subsets of /3" are split, The proof is quite different from Shore's and is more in the spirit of Owings' splitting theorem [11] for metarecursion theory. A second way of approaching this problem is to assume that the B-r.e. set we are trying to split is regular and to later examine which fl-r.e, degrees contain such sets, Another problem arises here in trying to generalize the proof from * The results of this paper arc from the author's Ph.D~ thesis written at M.I.T. in 1978. I would like to thank my th~is advisor. Professor Gerald Sacks. for his encouragement and guidance during the preparation of this wx~rk.
Proceedings 35th Annual Symposium on Foundations of Computer Science
We present a polynomial time online learning algorithm that learns any discretized geometric conc... more We present a polynomial time online learning algorithm that learns any discretized geometric concept generated from any number of halfspaces with any number of known (to the learner) slopes in a constant dimensional space. In particular, our algorithm learns (from equivalence queries only) unions of discretized axis-parallel rectangles in a constant dimensional space in polynomial time. The algorithm also runs in polynomial time an 1 if the teacher lies on 1 counterexamples. We then show a PAC-learning algorithm for the above discretized geometric concept when the example oracle lies on the labels of the examples with a fixed probability p 5 3that runs in polynomial time also with r. We use these methods, as well as a bounded version of the finite injury priority method, to construct algorithms f o r learning several classes of rectangles. In particular we design eficient algorithms for learning several classes of unions of discretized axis-parallel rectangles in either arbitrary dimensional spaces or constant dimensional spaces.
We define and construct efficient depth-universal and almostsize-universal quantum circuits. Such... more We define and construct efficient depth-universal and almostsize-universal quantum circuits. Such circuits can be viewed as generalpurpose simulators for central quantum circuit classes and used to capture the computational power of the simulated class. For depth we construct universal circuits whose depth is the same order as the circuits being simulated. For size, there is a log factor blow-up in the universal circuits constructed here which is nearly optimal for polynomial size circuits. {(dbera|homer)}@cs.bu.edu.
Abstract. We study the problem of finding a hidden code k in the domain (1,..., my* in the presen... more Abstract. We study the problem of finding a hidden code k in the domain (1,..., my* in the presence of an oracle which, for any z in the domain, answers a pair of numbers a(z, k) and b(z, k) such that a(z, k) is the number of components coinciding in z and k and, b(z, k) is the sum of ...
We demonstrate differences between reducibilities and corresponding completeness notions for nond... more We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomial-time bounded (even logarithmic-space bounded) reducibilities turn out to be different for any class containing NE. For space classes the completeness notions under logspace reducibilities can be separated for any class properly containing LOGSPACE. Key observation in obtaining the separations is the honesty property of reductions, which was recently observed to hold for the time classes and can be shown to hold for space classes.
We consider under the assumption PINP questions concerning the structure of the lattice of NP set... more We consider under the assumption PINP questions concerning the structure of the lattice of NP sets together with the sublattice P. We show that two questions which are slightly more complex than the known splitting properties of this lattice cannot be settled by arguments which relativize. The two questions which we consider are whether every infinite NP set contains an infinite P subset and whether there exists an NP-simple set. We construct several oracles, all of which make P I NP, and which in addition make the above-mentioned statements either true or false. In particular we give a positive answer to the question, raised by Bennett and Gill (1981), whether an oracle B exists making PBINPt and such that every infinite set in NPB has an infinite subset in PB. The constructions of the oracles are finite injury priority arguments.
The existence of minimal degrees is investigated for several polynomial reducibilities. It is sho... more The existence of minimal degrees is investigated for several polynomial reducibilities. It is shown that no set has minimal degree with respect to polynomial many-one or Turing reducibility. This extends a result of Ladner in which only recursive sets are considered. A polynomial reducibility ≤<supscrpt><italic>h</italic></supscrpt><subscrpt><italic>T</italic></subscrpt> is defined. This reducibility is a strengthening of polynomial Turing reducibility, and its properties relate to the P = ? NP question. For this new reducibility, a set of minimal degree is constructed under the assumption that P = NP. However, the set constructed is nonrecursive, and it is shown that no recursive set is of minimal ≤ <supscrpt><italic>h</italic></supscrpt><subscrpt><italic>T</italic></subscrpt> degree.
Journal of Parallel and Distributed Computing, 1997
We develop and experiment with a new parallel algorithm to approximate the maximum weight cut in ... more We develop and experiment with a new parallel algorithm to approximate the maximum weight cut in a weighted undirected graph. Our implementation starts with the recent (serial) algorithm of Goemans and Williamson for this problem. We consider several different versions of this algorithm, varying the interior-point part of the algorithm in order to optimize the parallel efficiency of our method. Our work aims for an efficient, practical formulation of the algorithm with closeto-optimal parallelization. We analyze our parallel algorithm in the LogP model and predict linear speedup for a wide range of the parameters. We have implemented the algorithm using the message passing interface (MPI) and run it on several parallel machines. In particular, we present performance measurements on the IBM SP2, the Connection Machine CM5, and a cluster of workstations. We observe that the measured speedups are predicted well by our analysis in the LogP model. Finally, we test our implementation on several large graphs (up to 13,000 vertices), particularly on large instances of the Ising model.
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