On the Complexity of the Orbit Problem

We consider higher-dimensional versions of Kannan and Lipton's Orbit Problem---determining whether a target vector space V may be reached from a starting point x under repeated applications of a linear transformation A. Answering two questions posed by Kannan and Lipton in the 1980s, we show that when V has dimension one, this problem is solvable in polynomial time, and when V has dimension two or three, the problem is in NP^{RP}

A linear time algorithm for the orbit problem over cyclic groups

The orbit problem is at the heart of symmetry reduction methods for model checking concurrent systems. It asks whether two given configurations in a concurrent system (represented as finite strings over some finite alphabet) are in the same orbit with respect to a given finite permutation group (represented by their generators) acting on this set of configurations by permuting indices. It is known that the problem is in general as hard as the graph isomorphism problem, whose precise complexity (whether it is solvable in polynomial-time) is a long-standing open problem. In this paper, we consider the restriction of the orbit problem when the permutation group is cyclic (i.e. generated by a single permutation), an important restriction of the problem. It is known that this subproblem is solvable in polynomial-time. Our main result is a linear-time algorithm for this subproblem.Comment: Accepted in Acta Informatica in Nov 201

No occurrence obstructions in geometric complexity theory

The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VP_{ws} and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of GL_{n^2}(C), which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via multiplicity obstructions as proposed by Mulmuley and Sohoni.Comment: Substantial revision. This version contains an overview of the proof of the main result. Added material on the model of power sums. Theorem 4.14 in the old version, which had a complicated proof, became the easy Theorem 5.4. To appear in the Journal of the AM

Prediction of the aerodynamic performance of re-usable single stage to orbit vehicles

Re-usable single stage to orbit launch vehicles promise to reduce the cost of access to space, but their success will be particularly reliant on accurate modelling of their aero-thermodynamic characteristics. Non-equilibrium effects due to the rarefaction of the gas in the atmosphere are important at the very high altitudes at which lifting R-SSTO configurations will experience their greatest thermal load during re-entry. Current limitations in modelling the behaviour of the gas and hence in capturing these effects have a strong impact on the accuracy with which the thermal and aerodynamic loading on the surface of the vehicle can be predicted during this design-critical flight regime. The problem is most apparent in the presence of strong shock interactions, and this is likely to exacerbate the problem of aerodynamic characterisation of re-usable single stage to orbit vehicles, especially given design pressures towards increased geometric complexity compared to historical spacecraft designs, and hence the complexity of the shock structures that the vehicle will produce in high-speed flight. The development of this class of vehicles will thus very likely be paced by the development of the specialised modelling tools that will be required to account fully for the properties of the gas at the high speeds and altitudes that are characteristic of their re-entry into the atmosphere of the earth

Avoiding Abelian powers in binary words with bounded Abelian complexity

The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden's theorem, they proved that if a word avoids Abelian $k$-powers for some integer $k$, then its Abelian complexity is unbounded. This suggests the following question: How frequently do Abelian $k$-powers occur in a word having bounded Abelian complexity? In particular, does every uniformly recurrent word having bounded Abelian complexity begin in an Abelian $k$-power? While this is true for various classes of uniformly recurrent words, including for example the class of all Sturmian words, in this paper we show the existence of uniformly recurrent binary words, having bounded Abelian complexity, which admit an infinite number of suffixes which do not begin in an Abelian square. We also show that the shift orbit closure of any infinite binary overlap-free word contains a word which avoids Abelian cubes in the beginning. We also consider the effect of morphisms on Abelian complexity and show that the morphic image of a word having bounded Abelian complexity has bounded Abelian complexity. Finally, we give an open problem on avoidability of Abelian squares in infinite binary words and show that it is equivalent to a well-known open problem of Pirillo-Varricchio and Halbeisen-Hungerb\"uhler.Comment: 16 pages, submitte

Multireference Alignment is Easier with an Aperiodic Translation Distribution

In the multireference alignment model, a signal is observed by the action of a random circular translation and the addition of Gaussian noise. The goal is to recover the signal's orbit by accessing multiple independent observations. Of particular interest is the sample complexity, i.e., the number of observations/samples needed in terms of the signal-to-noise ratio (the signal energy divided by the noise variance) in order to drive the mean-square error (MSE) to zero. Previous work showed that if the translations are drawn from the uniform distribution, then, in the low SNR regime, the sample complexity of the problem scales as $\omega(1/\text{SNR}^3)$. In this work, using a generalization of the Chapman--Robbins bound for orbits and expansions of the $\chi^2$ divergence at low SNR, we show that in the same regime the sample complexity for any aperiodic translation distribution scales as $\omega(1/\text{SNR}^2)$. This rate is achieved by a simple spectral algorithm. We propose two additional algorithms based on non-convex optimization and expectation-maximization. We also draw a connection between the multireference alignment problem and the spiked covariance model

Activity Planning for a Lunar Orbital Mission

This paper describes a challenging, real-world planning problem within the context of a NASA mission called LADEE (Lunar Atmospheric Dust Environment Explorer). LADEEs science phase was performed in an equatorial, retrograde orbit around the Moon. The science observations were constrained with respect to key points in the spacecrafts orbit. We present the approach taken to reduce the complexity of the activity planning task in order to effectively perform it within the time pressures imposed by the mission requirements. One key aspect of this approach is the design of the activity planning process based on principles of problem decomposition and planning abstraction levels. The second key aspect is the mixed-initiative system developed for this task, called LASS (LADEE Activity Scheduling System). The primary challenge for LASS was representing and managing the orbit-based science constraints, given their dynamic nature due to the continually updated orbit determination solution

Estimation under group actions: recovering orbits from invariants

Motivated by geometric problems in signal processing, computer vision, and structural biology, we study a class of orbit recovery problems where we observe very noisy copies of an unknown signal, each acted upon by a random element of some group (such as Z/p or SO(3)). The goal is to recover the orbit of the signal under the group action in the high-noise regime. This generalizes problems of interest such as multi-reference alignment (MRA) and the reconstruction problem in cryo-electron microscopy (cryo-EM). We obtain matching lower and upper bounds on the sample complexity of these problems in high generality, showing that the statistical difficulty is intricately determined by the invariant theory of the underlying symmetry group. In particular, we determine that for cryo-EM with noise variance $\sigma^2$ and uniform viewing directions, the number of samples required scales as $\sigma^6$. We match this bound with a novel algorithm for ab initio reconstruction in cryo-EM, based on invariant features of degree at most 3. We further discuss how to recover multiple molecular structures from heterogeneous cryo-EM samples.Comment: 54 pages. This version contains a number of new result

Space debris cataloging of GEO objects by using Meta-Heuristic methods

Currently several thousands of objects are being tracked in the Medium Earth Orbit (MEO) and Geosynchronous Earth Orbit (GEO) regions through optical means. The problem faced in this framework is that of Multiple Target Tracking (MTT). The MTT problem becomes an NP-hard combinatorial optimization problem as soon as its dimension S becomes S ≥ 3. In regions with a high density of objects the MTT problem will have to have this dimension in order to avoid ambiguous solutions. With the advent of improved sensors and a eightened interest in the problem of space debris, it is expected that the number of tracked objects will grow by an order of magnitude in the near future. This research aims to identify an algorithm capable of addressing the problem of space debris cataloging in the MEO and GEO regions, in particular for highly dense regions, without possessing a restrictive computational complexity. In an attempt to find an approximate solution of sufficient quality several Population Based Meta Heuristic (PBMH) algorithms are implemented and tested on simulated optical measurements. In addition to this, a novel way of orbit determination is presented which is based on an existing S = 2 tracklet association method. These first results show promise as one of the tested algorithms (the Elitist Genetic Algorithm (EGA)) consistently displays the desired behavior of finding good approximate solutions before reaching the optimum. Furthermore, the results suggest that the algorithm has a polynomial time complexity when finding approximate solutions. The algorithm is also applied to real observations, where it also performs as desired

Geometric complexity theory and matrix powering

Valiant's famous determinant versus permanent problem is the flagship problem in algebraic complexity theory. Mulmuley and Sohoni (Siam J Comput 2001, 2008) introduced geometric complexity theory, an approach to study this and related problems via algebraic geometry and representation theory. Their approach works by multiplying the permanent polynomial with a high power of a linear form (a process called padding) and then comparing the orbit closures of the determinant and the padded permanent. This padding was recently used heavily to show no-go results for the method of shifted partial derivatives (Efremenko, Landsberg, Schenck, Weyman, 2016) and for geometric complexity theory (Ikenmeyer Panova, FOCS 2016 and B\"urgisser, Ikenmeyer Panova, FOCS 2016). Following a classical homogenization result of Nisan (STOC 1991) we replace the determinant in geometric complexity theory with the trace of a variable matrix power. This gives an equivalent but much cleaner homogeneous formulation of geometric complexity theory in which the padding is removed. This radically changes the representation theoretic questions involved to prove complexity lower bounds. We prove that in this homogeneous formulation there are no orbit occurrence obstructions that prove even superlinear lower bounds on the complexity of the permanent. This is the first no-go result in geometric complexity theory that rules out superlinear lower bounds in some model. Interestingly---in contrast to the determinant---the trace of a variable matrix power is not uniquely determined by its stabilizer