Approximating the Permanent with Fractional Belief Propagation

We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze the Belief Propagation (BP) approach and its Fractional Belief Propagation (FBP) generalization for computing the permanent of a non-negative matrix. Known bounds and conjectures are verified in experiments, and some new theoretical relations, bounds and conjectures are proposed. The Fractional Free Energy (FFE) functional is parameterized by a scalar parameter $\gamma\in[-1;1]$, where $\gamma=-1$ corresponds to the BP limit and $\gamma=1$ corresponds to the exclusion principle (but ignoring perfect matching constraints) Mean-Field (MF) limit. FFE shows monotonicity and continuity with respect to $\gamma$. For every non-negative matrix, we define its special value $\gamma_*\in[-1;0]$ to be the $\gamma$ for which the minimum of the $\gamma$-parameterized FFE functional is equal to the permanent of the matrix, where the lower and upper bounds of the $\gamma$-interval corresponds to respective bounds for the permanent. Our experimental analysis suggests that the distribution of $\gamma_*$ varies for different ensembles but $\gamma_*$ always lies within the $[-1;-1/2]$ interval. Moreover, for all ensembles considered the behavior of $\gamma_*$ is highly distinctive, offering an emprirical practical guidance for estimating permanents of non-negative matrices via the FFE approach.Comment: 42 pages, 14 figure

Approximating the Permanent of a Random Matrix with Vanishing Mean

We show an algorithm for computing the permanent of a random matrix with vanishing mean in quasi-polynomial time. Among special cases are the Gaussian, and biased-Bernoulli random matrices with mean 1/lnln(n)^{1/8}. In addition, we can compute the permanent of a random matrix with mean 1/poly(ln(n)) in time 2^{O(n^{\eps})} for any small constant \eps>0. Our algorithm counters the intuition that the permanent is hard because of the "sign problem" - namely the interference between entries of a matrix with different signs. A major open question then remains whether one can provide an efficient algorithm for random matrices of mean 1/poly(n), whose conjectured #P-hardness is one of the baseline assumptions of the BosonSampling paradigm

Scattering of slow-light gap solitons with charges in a two-level medium

The Maxwell-Bloch system describes a quantum two-level medium interacting with a classical electromagnetic field by mediation of the the population density. This population density variation is a purely quantum effect which is actually at the very origin of nonlinearity. The resulting nonlinear coupling possesses particularly interesting consequences at the resonance (when the frequency of the excitation is close to the transition frequency of the two-level medium) as e.g. slow-light gap solitons that result from the nonlinear instability of the evanescent wave at the boundary. As nonlinearity couples the different polarizations of the electromagnetic field, the slow-light gap soliton is shown to experience effective scattering whith charges in the medium, allowing it for instance to be trapped or reflected. This scattering process is understood qualitatively as being governed by a nonlinear Schroedinger model in an external potential related to the charges (the electrostatic permanent background component of the field).Comment: RevTex, 14 pages with 5 figures, to appear in J. Phys. A: Math. Theo

Rules of Thumb in Life-Cycle Saving Decisions

We analyse life-cycle saving decisions when households use simple heuristics, or rules of thumb, rather than solve the underlying intertemporal optimization problem. We simulate life-cycle saving decisions using three simple rules and compute utility losses relative to the solution of the optimization problem. Our simulations suggest that utility losses induced by following simple decision rules are relatively low. Moreover, the two main saving motives re ected by the canonical life-cycle model { long-run consumption smoothing and short-run insurance against income shocks { can be addressed quite well by saving rules that do not require computationally demanding tasks such as backward induction

The Total Takings Myth

For almost thirty-five years, the U.S. Supreme Court has attempted to carve out a total takings doctrine within its regulatory takings jurisprudence. Most regulatory takings claims are evaluated under the “ad hoc” threefactor test first articulated in Penn Central Transportation Co. v. City of New York. Exceedingly few of these claims are successful. But the Court has identified certain categories of government actions that are compensable takings per se, otherwise known as total takings. This began in 1982 with Loretto v. Teleprompter Manhattan CATV Corp., where the Court held that a land use ordinance requiring a landowner to endure a permanent physical occupation of a portion of her property is always a compensable taking. Ten years later, in Lucas v. South Carolina Coastal Council, the Court held that a land use restriction depriving an owner of all economically viable use of her property is also compensable per se. Finally, in 2015, in Horne v. Department of Agriculture, the Court extended its total takings jurisprudence to personal property, announcing that the government appropriation of personal property is a per se compensable taking. Although the Court has had more than three decades to articulate theoretical justifications for its total takings jurisprudence and to provide guidance for lower courts in determining when a regulation constitutes a total taking, it has failed to do so. This failure reflects the underlying reality that the total takings doctrine is a myth. More particularly, the categories that the Court has identified as constituting total takings are analytically incoherent, and the terms the Court has used to demarcate total takings from regulations that are not per se compensable cannot be applied in the real world. As a result, lower courts struggle to apply the total takings doctrine and the case law remains in utter disarray. In fact, lower courts have resorted to creating “shadow” total takings doctrines that rely on obvious distortions of the plain meaning of outcome-determinative terms and deflect attention from the fundamental question of whether compensation is warranted. This Article argues that the Court’s attempt to create a total takings doctrine has failed, and that the Court should repudiate it. It demonstrates that the Court’s initial total takings opinions were conceptually incoherent and woefully undertheorized. And it shows that attempts by lower courts to rehabilitate the doctrine by crystallizing the bright-line rules through careful and consistent application were doomed to, and did, fail. This Article also explains why the entire enterprise was misguided from the start. Although bright-line rules have their place, it is not in the heart of regulatory takings doctrine, which is premised on concerns for fairness and justice in distributing the burdens of land use regulation. Last term, the Court had a perfect opportunity to begin the process of repudiating the total takings myth. Murr v. Wisconsin was a run-of-the-mill regulatory takings case masquerading as a Lucas-type total takings claim, and it presented the Court with a vehicle to either remedy the central doctrinal incoherence of Lucas’s bright-line rule or to overrule Lucas and turn its attention to the much needed task of clarifying and refining the Penn Central test. Instead, by offering a new multifactored test in a sort of regulatory takings “step zero,” the Court in Murr merely exacerbated the core flaws of the Lucas bright-line rule. Now, more than ever, it is imperative that the Court recognize and begin to dismantle the total takings myth

Web Note No. 9

In 2008 the Alaska Legislature passed and the governor signed into law a bill requiring the Office of Management and Budget (OMB) to prepare an annual state fiscal plan projecting state spending for 10 years and identifying the revenue sources to pay for that spending. One objective of the law was to get government and the general public thinking, discussing, and planning for the long-term fiscal health of the state in light of declining oil production. These plans have not attracted the attention they deserve. In this Web Note we review the most recent fiscal year 2012 10-year plan and offer suggestions for improvement.Northrim Bank