A Note on the Energy Release Rate in Quasi-Static Elastic Crack Propagation

This paper considers analytical issues associated with the notion of the energy release rate in quasi-static elastic crack propagation

Machine learning: statistical physics based theory and smart industry applications

The increasing computational power and the availability of data have made it possible to train ever-bigger artificial neural networks. These so-called deep neural networks have been used for impressive applications, like advanced driver assistance and support in medical diagnoses. However, various vulnerabilities have been revealed and there are many open questions concerning the workings of neural networks. Theoretical analyses are therefore essential for further progress. One current question is: why is it that networks with Rectified Linear Unit (ReLU) activation seemingly perform better than networks with sigmoidal activation?We contribute to the answer to this question by comparing ReLU networks with sigmoidal networks in diverse theoretical learning scenarios. In contrast to analysing specific datasets, we use a theoretical modelling using methods from statistical physics. They give the typical learning behaviour for chosen model scenarios. We analyse both the learning behaviour on a fixed dataset and on a data stream in the presence of a changing task. The emphasis is on the analysis of the network’s transition to a state wherein specific concepts have been learnt. We find significant benefits of ReLU networks: they exhibit continuous increases of their performance and adapt more quickly to changing tasks.In the second part of the thesis we treat applications of machine learning: we design a quick quality control method for material in a production line and study the relationship with product faults. Furthermore, we introduce a methodology for the interpretable classification of time series data

Do Prices Coordinate Markets?

Walrasian equilibrium prices can be said to coordinate markets: They support a welfare optimal allocation in which each buyer is buying bundle of goods that is individually most preferred. However, this clean story has two caveats. First, the prices alone are not sufficient to coordinate the market, and buyers may need to select among their most preferred bundles in a coordinated way to find a feasible allocation. Second, we don't in practice expect to encounter exact equilibrium prices tailored to the market, but instead only approximate prices, somehow encoding "distributional" information about the market. How well do prices work to coordinate markets when tie-breaking is not coordinated, and they encode only distributional information? We answer this question. First, we provide a genericity condition such that for buyers with Matroid Based Valuations, overdemand with respect to equilibrium prices is at most 1, independent of the supply of goods, even when tie-breaking is done in an uncoordinated fashion. Second, we provide learning-theoretic results that show that such prices are robust to changing the buyers in the market, so long as all buyers are sampled from the same (unknown) distribution

Comparing spatial features of urban housing markets:

Various location specific attributes contribute to the spatial dynamics of housing markets. This effect may partly be of a qualitative and discontinuous nature, which causes market segmentation into submarkets. The question however is, whether the most relevant partitioning criteria is directly related to the transaction price of to other, socioeconomic, demographic and physical features of the location. Two neural network techniques are used for analysing statistical house price data from Amsterdam and Helsinki. The analytic hierarchy process is used as a supporting technique. With these techniques it is possible to analyse various dimensions of housing submarket formation. The findings show that, while the price and demand factors have increased in importance, supply factors still prevail as key criteria in both cases. The outcome also indicates that the housing market structure of Amsterdam is more fragmented than that of Helsinki, and that the main discriminating housing market features, and the ways they have changed in time, are somewhat different

Phase transitions in Vector Quantization

We study Winner-Takes-All and rank based Vector Quantization along the lines of the statistical physics of off-line learning. Typical behavior of the system is obtained within a model where high-dimensional training data are drawn from a mixture of Gaussians. The analysis becomes exact in the simplifying limit of high training temperature. Our main findings concern the existence of phase transitions, i.e. a critical or discontinuous dependence of VQ performance on the training set size. We show how the nature and properties of the transition depend on the number of prototypes and the control parameter of rank based cost functions.