Stable comparison of multidimensional persistent homology groups with torsion

The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance d_T that represents a possible solution to this problem. Indeed, d_T is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the new pseudo-distance with respect to the change of the filtering function, expressed both with respect to the max-norm and to the natural pseudo-distance between topological spaces endowed with vector-valued filtering functions. Furthermore, we prove a result showing the relationship between d_T and the matching distance in the 1-dimensional case, when the homology coefficients are taken in a field and hence the comparison can be made

Stable comparison of multidimensional persistent homology groups with torsion

The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance d_T that represents a possible solution to this problem. Indeed, d_T is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the new pseudo-distance with respect to the change of the filtering function, expressed both with respect to the max-norm and to the natural pseudo-distance between topological spaces endowed with vector-valued filtering functions. Furthermore, we prove a result showing the relationship between d_T and the matching distance in the 1-dimensional case, when the homology coefficients are taken in a field and hence the comparison can be made.Comment: 10 pages, 3 figure

Revisiting Public Debt and Inflation: Fiscal Implications of an Independent Central Banker

The mainstream literature on monetary policy games under output persistence posits that: a) monetary regimes do not affect real variables in the steady state; b) optimal institutional design should entirely remove the inflation bias. We show that neither result necessarily holds if output persistence originates from debt dynamics and distortionary taxation. First, monetary delegation induces a strategic use of debt policy affecting steady-state distortions. Second, the reduction of such distortions may require monetary institutions that tolerate an inflation rate above the socially optimal level.

An empirical investigation of the relationship between inequality and growth

This paper studies the correlation between inequality, measured by the Gini coefficent of incomes, and the growth rate of per capita GDP in a panel of countries between the late 1950s and late 1990s. Inequality Granger causes growth with a negative coefficient, while growth Granger causes inequality with a positive sign. Quantitatively, the former effect appears much larger than the latter. Once I allow for the effect to differ between rich and poor countries interesting differences emerge. While lagged inequality appears positively correlated with growth in the subgroup of rich countries, in poor countries besides a negative and significant effect of lagged inequality on growth there is a negative and significant effect of lagged growth on inequalitygrowth; inequality; panel; GMM; Granger causality

Dynamic Seigniorage Models Revisited. Should Fiscal Flexibility and Conservative Central Bankers Go Together?

This paper presents a dynamic seigniorage model where excessive debt levels persist in steady state, causing a permanent inflation bias. Discretionary monetary responses to shocks are too interventionist because they do not take into account the role of debt policy, which spreads part of the adjustment onto future periods. Institutional design should contemplate the appointment of weight-conservative central bankers. The central bank preferences should be more conservative the more the government is willing to delay the adjustment of expenditures following a supply shock. The combination of fiscal intervention and a zero inflation rule describes how members of a monetary union might react to asymmetric shocks. The costs of this regime are negligible if the discount factor is small and seigniorage losses are limited.

Optimal Simple Monetary and Fiscal Rules under Limited Asset Market Participation

The combination of limited asset market participation and consumption habits generates indeterminacy for empirically plausible calibrations of a business cycle model characterized by price and nominal wage rigidities. Equilibrium determinacy is restored by demand management policies based on simple fiscal rules. In this regard, fiscal control of nominal income growth is particularly effective. In addition the complementarity between the Taylor rule and the fiscal feedback on nominal income growth produces relatively large welfare gains, limiting both aggregate and intragroup volatilities

Invariance properties of the multidimensional matching distance in Persistent Topology and Homology

Persistent Topology studies topological features of shapes by analyzing the lower level sets of suitable functions, called filtering functions, and encoding the arising information in a parameterized version of the Betti numbers, i.e. the ranks of persistent homology groups. Initially introduced by considering real-valued filtering functions, Persistent Topology has been subsequently generalized to a multidimensional setting, i.e. to the case of $\R^n$-valued filtering functions, leading to studying the ranks of multidimensional homology groups. In particular, a multidimensional matching distance has been defined, in order to compare these ranks. The definition of the multidimensional matching distance is based on foliating the domain of the ranks of multidimensional homology groups by a collection of half-planes, and hence it formally depends on a subset of $\R^n\times\R^n$ inducing a parameterization of these half-planes. It happens that it is possible to choose this subset in an infinite number of different ways. In this paper we show that the multidimensional matching distance is actually invariant with respect to such a choice.Comment: 14 pages, 2 figure