147 research outputs found
Divergent estimation error in portfolio optimization and in linear regression
The problem of estimation error in portfolio optimization is discussed, in
the limit where the portfolio size N and the sample size T go to infinity such
that their ratio is fixed. The estimation error strongly depends on the ratio
N/T and diverges for a critical value of this parameter. This divergence is the
manifestation of an algorithmic phase transition, it is accompanied by a number
of critical phenomena, and displays universality. As the structure of a large
number of multidimensional regression and modelling problems is very similar to
portfolio optimization, the scope of the above observations extends far beyond
finance, and covers a large number of problems in operations research, machine
learning, bioinformatics, medical science, economics, and technology.Comment: 5 pages, 2 figures, Statphys 23 Conference Proceedin
Large random correlations in individual mean field spin glass samples
We argue that complex systems must possess long range correlations and
illustrate this idea on the example of the mean field spin glass model. Defined
on the complete graph, this model has no genuine concept of distance, but the
long range character of correlations is translated into a broad distribution of
the spin-spin correlation coefficients for almost all realizations of the
random couplings. When we sample the whole phase space we find that this
distribution is so broad indeed that at low temperatures it essentially becomes
uniform, with all possible correlation values appearing with the same
probability. The distribution of correlations inside a single phase space
valley is also studied and found to be much narrower.Comment: Added a few references and a comment phras
The effect of social balance on social fragmentation
With the availability of cell phones, internet, social media etc. the interconnectedness of people within most societies has increased drastically over the past three decades. Across the same timespan, we are observing the phenomenon of increasing levels of fragmentation in society into relatively small and isolated groups that have been termed filter bubbles, or echo chambers. These pose a number of threats to open societies, in particular, a radicalisation in political, social or cultural issues, and a limited access to facts. In this paper we show that these two phenomena might be tightly related. We study a simple stochastic co-evolutionary model of a society of interacting people. People are not only able to update their opinions within their social context, but can also update their social links from collaborative to hostile, and vice versa. The latter is implemented such that social balance is realised. We find that there exists a critical level of interconnectedness, above which society fragments into small sub-communities that are positively linked within and hostile towards other groups. We argue that the existence of a critical communication density is a universal phenomenon in all societies that exhibit social balance. The necessity arises from the underlying mathematical structure of a phase transition phenomenon that is known from the theory of a kind of disordered magnets called spin glasses. We discuss the consequences of this phase transition for social fragmentation in society
Against Chaos in Temperature in Mean-Field Spin-Glass Models
We study the problem of chaos in temperature in some mean-field spin-glass
models by means of a replica computation over a model of coupled systems. We
propose a set of solutions of the saddle point equations which are
intrinsically non-chaotic and solve a general problem regarding the consistency
of their structure. These solutions are relevant in the case of uncoupled
systems too, therefore they imply a non-trivial overlap distribution
between systems at different temperatures. The existence of such
solutions is checked to fifth order in an expansion near the critical
temperature through highly non-trivial cancellations, while it is proved that a
dangerous set of such cancellations holds exactly at all orders in the
Sherrington-Kirkpatrick (SK) model. The SK model with soft-spin distribution is
also considered obtaining analogous results. Previous analytical results are
discussed.Comment: 20 pages, submitted to J.Phys.
Regularizing Portfolio Optimization
The optimization of large portfolios displays an inherent instability to
estimation error. This poses a fundamental problem, because solutions that are
not stable under sample fluctuations may look optimal for a given sample, but
are, in effect, very far from optimal with respect to the average risk. In this
paper, we approach the problem from the point of view of statistical learning
theory. The occurrence of the instability is intimately related to over-fitting
which can be avoided using known regularization methods. We show how
regularized portfolio optimization with the expected shortfall as a risk
measure is related to support vector regression. The budget constraint dictates
a modification. We present the resulting optimization problem and discuss the
solution. The L2 norm of the weight vector is used as a regularizer, which
corresponds to a diversification "pressure". This means that diversification,
besides counteracting downward fluctuations in some assets by upward
fluctuations in others, is also crucial because it improves the stability of
the solution. The approach we provide here allows for the simultaneous
treatment of optimization and diversification in one framework that enables the
investor to trade-off between the two, depending on the size of the available
data set
One-step replica symmetry breaking solution of the quadrupolar glass model
We consider the quadrupolar glass model with infinite-range random
interaction. Introducing a simple one-step replica symmetry breaking ansatz we
investigate the para-glass continuous (discontinuous) transition which occurs
below (above) a critical value of the quadrupole dimension m*. By using a
mean-field approximation we study the stability of the one-step replica
symmetry breaking solution and show that for m>m* there are two transitions.
The thermodynamic transition is discontinuous but there is no latent heat. At a
higher temperature we find the dynamical or glass transition temperature and
the corresponding discontinuous jump of the order parameter.Comment: 10 pages, 3 figure
Evidences Against Temperature Chaos in Mean Field and Realistic Spin Glasses
We discuss temperature chaos in mean field and realistic 3D spin glasses. Our
numerical simulations show no trace of a temperature chaotic behavior for the
system sizes considered. We discuss the experimental and theoretical
implications of these findings.Comment: 4 pages in aps format. 6 .ps figures. It is better to print the paper
in colou
Large Deviation Property of Free Energy in p-Body Sherrington-Kirkpatrick Model
Cumulant generating function phi(n) and rate function Sigma(f) of the free
energy is evaluated in p-body Sherrington-Kirkpatrick model by using the
replica method with the replica number n finite. From a perturbational
argument, we show that the cumulant generating function is constant in the
vicinity of n = 0. On the other hand, with the help of two analytic properties
of phi(n), the behavior of phi(n) is derived again. However this is also shown
to be broken at a finite value of n, which gives a characteristic value in the
rate function near the thermodynamic value of the free energy. Through the
continuation of phi(n) as a function of n, we find out a way to derive the 1RSB
solution at least in this model, which is to fix the RS solution to be a
monotone increasing function.Comment: 7 pages, 5 figures. accepted for publication in J.Phs.Soc.Jp
DeepWalk: Online Learning of Social Representations
We present DeepWalk, a novel approach for learning latent representations of
vertices in a network. These latent representations encode social relations in
a continuous vector space, which is easily exploited by statistical models.
DeepWalk generalizes recent advancements in language modeling and unsupervised
feature learning (or deep learning) from sequences of words to graphs. DeepWalk
uses local information obtained from truncated random walks to learn latent
representations by treating walks as the equivalent of sentences. We
demonstrate DeepWalk's latent representations on several multi-label network
classification tasks for social networks such as BlogCatalog, Flickr, and
YouTube. Our results show that DeepWalk outperforms challenging baselines which
are allowed a global view of the network, especially in the presence of missing
information. DeepWalk's representations can provide scores up to 10%
higher than competing methods when labeled data is sparse. In some experiments,
DeepWalk's representations are able to outperform all baseline methods while
using 60% less training data. DeepWalk is also scalable. It is an online
learning algorithm which builds useful incremental results, and is trivially
parallelizable. These qualities make it suitable for a broad class of real
world applications such as network classification, and anomaly detection.Comment: 10 pages, 5 figures, 4 table
Chaos and Universality in a Four-Dimensional Spin Glass
We present a finite size scaling analysis of Monte Carlo simulation results
on a four dimensional Ising spin glass. We study chaos with both coupling and
temperature perturbations, and find the same chaos exponent in each case. Chaos
is investigated both at the critical temperature and below where it seems to be
more efficient (larger exponent). Dimension four seems to be above the critical
dimension where chaos with temperature is no more present in the critical
region. Our results are consistent with the Gaussian and bimodal coupling
distributions being in the same universality class.Comment: 11 pages, including 6 postscript figures. Latex with revtex macro
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