1,159 research outputs found
Quantum ground state isoperimetric inequalities for the energy spectrum of local Hamiltonians
We investigate the relationship between the energy spectrum of a local
Hamiltonian and the geometric properties of its ground state. By generalizing a
standard framework from the analysis of Markov chains to arbitrary
(non-stoquastic) Hamiltonians we are naturally led to see that the spectral gap
can always be upper bounded by an isoperimetric ratio that depends only on the
ground state probability distribution and the range of the terms in the
Hamiltonian, but not on any other details of the interaction couplings. This
means that for a given probability distribution the inequality constrains the
spectral gap of any local Hamiltonian with this distribution as its ground
state probability distribution in some basis (Eldar and Harrow derived a
similar result in order to characterize the output of low-depth quantum
circuits). Going further, we relate the Hilbert space localization properties
of the ground state to higher energy eigenvalues by showing that the presence
of k strongly localized ground state modes (i.e. clusters of probability, or
subsets with small expansion) in Hilbert space implies the presence of k energy
eigenvalues that are close to the ground state energy. Our results suggest that
quantum adiabatic optimization using local Hamiltonians will inevitably
encounter small spectral gaps when attempting to prepare ground states
corresponding to multi-modal probability distributions with strongly localized
modes, and this problem cannot necessarily be alleviated with the inclusion of
non-stoquastic couplings
Absorption Time of the Moran Process
The Moran process models the spread of mutations in populations on graphs. We
investigate the absorption time of the process, which is the time taken for a
mutation introduced at a randomly chosen vertex to either spread to the whole
population, or to become extinct. It is known that the expected absorption time
for an advantageous mutation is O(n^4) on an n-vertex undirected graph, which
allows the behaviour of the process on undirected graphs to be analysed using
the Markov chain Monte Carlo method. We show that this does not extend to
directed graphs by exhibiting an infinite family of directed graphs for which
the expected absorption time is exponential in the number of vertices. However,
for regular directed graphs, we show that the expected absorption time is
Omega(n log n) and O(n^2). We exhibit families of graphs matching these bounds
and give improved bounds for other families of graphs, based on isoperimetric
number. Our results are obtained via stochastic dominations which we
demonstrate by establishing a coupling in a related continuous-time model. The
coupling also implies several natural domination results regarding the fixation
probability of the original (discrete-time) process, resolving a conjecture of
Shakarian, Roos and Johnson.Comment: minor change
Zero Shot Learning with the Isoperimetric Loss
We introduce the isoperimetric loss as a regularization criterion for
learning the map from a visual representation to a semantic embedding, to be
used to transfer knowledge to unknown classes in a zero-shot learning setting.
We use a pre-trained deep neural network model as a visual representation of
image data, a Word2Vec embedding of class labels, and linear maps between the
visual and semantic embedding spaces. However, the spaces themselves are not
linear, and we postulate the sample embedding to be populated by noisy samples
near otherwise smooth manifolds. We exploit the graph structure defined by the
sample points to regularize the estimates of the manifolds by inferring the
graph connectivity using a generalization of the isoperimetric inequalities
from Riemannian geometry to graphs. Surprisingly, this regularization alone,
paired with the simplest baseline model, outperforms the state-of-the-art among
fully automated methods in zero-shot learning benchmarks such as AwA and CUB.
This improvement is achieved solely by learning the structure of the underlying
spaces by imposing regularity.Comment: Accepted to AAAI-2
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