Static Data Structure Lower Bounds Imply Rigidity

We show that static data structure lower bounds in the group (linear) model imply semi-explicit lower bounds on matrix rigidity. In particular, we prove that an explicit lower bound of $t \geq \omega(\log^2 n)$ on the cell-probe complexity of linear data structures in the group model, even against arbitrarily small linear space $(s= (1+\varepsilon)n)$, would already imply a semi-explicit ($\bf P^{NP}\rm$) construction of rigid matrices with significantly better parameters than the current state of art (Alon, Panigrahy and Yekhanin, 2009). Our results further assert that polynomial ($t\geq n^{\delta}$) data structure lower bounds against near-optimal space, would imply super-linear circuit lower bounds for log-depth linear circuits (a four-decade open question). In the succinct space regime $(s=n+o(n))$, we show that any improvement on current cell-probe lower bounds in the linear model would also imply new rigidity bounds. Our results rely on a new connection between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak, 2006), and on a new reduction from worst-case to average-case rigidity, which is of independent interest

Equivalence of Systematic Linear Data Structures and Matrix Rigidity

Recently, Dvir, Golovnev, and Weinstein have shown that sufficiently strong lower bounds for linear data structures would imply new bounds for rigid matrices. However, their result utilizes an algorithm that requires an $NP$ oracle, and hence, the rigid matrices are not explicit. In this work, we derive an equivalence between rigidity and the systematic linear model of data structures. For the $n$-dimensional inner product problem with $m$ queries, we prove that lower bounds on the query time imply rigidity lower bounds for the query set itself. In particular, an explicit lower bound of $\omega\left(\frac{n}{r}\log m\right)$ for $r$ redundant storage bits would yield better rigidity parameters than the best bounds due to Alon, Panigrahy, and Yekhanin. We also prove a converse result, showing that rigid matrices directly correspond to hard query sets for the systematic linear model. As an application, we prove that the set of vectors obtained from rank one binary matrices is rigid with parameters matching the known results for explicit sets. This implies that the vector-matrix-vector problem requires query time $\Omega(n^{3/2}/r)$ for redundancy $r \geq \sqrt{n}$ in the systematic linear model, improving a result of Chakraborty, Kamma, and Larsen. Finally, we prove a cell probe lower bound for the vector-matrix-vector problem in the high error regime, improving a result of Chattopadhyay, Kouck\'{y}, Loff, and Mukhopadhyay.Comment: 23 pages, 1 tabl

A Framework for Identifying the Sources of Local-Currency Price Stability with an Empirical Application

The inertia of the local-currency prices of traded goods in the face of exchange-rate changes is a well-documented phenomenon in International Economics. This paper develops a framework for identifying the sources of local-currency price stability. The empirical approach exploits manufacturers’ and retailers’ first-order conditions in conjunction with detailed information on the frequency of price adjustments in response to exchange-rate changes, in order to quantify the relative importance of markup adjustment by manufacturers and retailers, local-cost non-traded components, and nominal price rigidities, in the incomplete transmission of exchange-rate changes to prices. The approach is applied to micro data from the beer market. We find that on average, 54.1% of the incomplete exchange rate pass-through is due to local non-traded costs; 33.7% to markup adjustment; and 12.2% to the existence of price adjustment costs.currency prices, exchange rates

Proof of the Riemannian Penrose Inequality with Charge for Multiple Black Holes

We present a proof of the Riemannian Penrose inequality with charge in the context of asymptotically flat initial data sets for the Einstein-Maxwell equations, having possibly multiple black holes with no charged matter outside the horizon, and satisfying the relevant dominant energy condition. The proof is based on a generalization of Hubert Bray's conformal flow of metrics adapted to this setting.Comment: 37 pages; final version; to appear in J. Differential Geo

A Structural Approach to Identifying the Sources of Local-Currency Price Stability

The inertia of the local-currency prices of traded goods in the face of exchange-rate changes is a well-documented phenomenon in International Economics. This paper develops a structural model to identify the sources of this local-currency price stability and applies it to micro data from the beer market. The empirical procedure exploits manufacturers’ and retailers’ first-order conditions in conjunction with detailed information on the frequency of price adjustments following exchange-rate changes to quantify the relative importance of local non-traded cost components, markup adjustment by manufacturers and retailers, and nominal price rigidities in the incomplete transmission of such changes to prices. We find that, on average, approximately 60% of the incomplete exchange rate pass-through is due to local non-traded costs; 8% to markup adjustment; 30% to the existence of own-brand price adjustment costs, and 1% to the indirect/strategic effect of such costs, though these results vary considerably across individual brands according to their market shares.

Price Rigidity and Strategic Uncertainty An Agent-based Approach

The phenomenon of infrequent price changes has troubled economists for decades. Intuitively one feels that for most price-setters there exists a range of inaction, i.e. a substantial measure of the states of the world, within which they do not wish to modify prevailing prices. However, basic economics tells us that when marginal costs change it is rational to change prices, too. Economists wishing to maintain rationality of price-setters resorted to fixed price adjustment costs as an explanation for price rigidity. In this paper we propose an alternative explanation, without recourse to any sort of physical adjustment cost, by putting strategic interaction into the center-stage of our analysis. Price-making is treated as a repeated oligopoly game. The traditional analysis of these games cannot pinpoint any equilibrium as a reasonable "solution" of the strategic situation. Thus there is genuine strategic uncertainty, a situation where decision-makers are uncertain of the strategies of other decision-makers. Hesitation may lead to inaction. To model this situation we follow the style of agent-based models, by modelling firms that change their pricing strategies following an evolutionary algorithm. Our results are promising. In addition to reproducing the known negative relationship between price rigidity and the level of general inflation, our model exhibits several features observed in real data. Moreover, most prices fall into the theoretical "range" without explicitly building this property into strategies.Agent-based modeling, Evolutionary algorithm, Price rigidity, Social learning, Strategic Uncertainty

Lower Bounds for Matrix Factorization

We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various incarnations in computer science; the most significant being in the context of lower bounds for algebraic circuits which compute linear transformations, matrix rigidity and data structure lower bounds. We first show, for every constant $d$, a deterministic construction in subexponential time of a family $\{M_n\}$ of $n \times n$ matrices which cannot be expressed as a product $M_n = A_1 \cdots A_d$ where the total sparsity of $A_1,\ldots,A_d$ is less than $n^{1+1/(2d)}$. In other words, any depth-$d$ linear circuit computing the linear transformation $M_n\cdot x$ has size at least $n^{1+\Omega(1/d)}$. This improves upon the prior best lower bounds for this problem, which are barely super-linear, and were obtained by a long line of research based on the study of super-concentrators (albeit at the cost of a blow up in the time required to construct these matrices). We then outline an approach for proving improved lower bounds through a certain derandomization problem, and use this approach to prove asymptotically optimal quadratic lower bounds for natural special cases, which generalize many of the common matrix decompositions