1,450 research outputs found
Large Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension
We review an approach which aims at studying discrete (pseudo-)manifolds in
dimension and called random tensor models. More specifically, we
insist on generalizing the two-dimensional notion of -angulations to higher
dimensions. To do so, we consider families of triangulations built out of
simplices with colored faces. Those simplices can be glued to form new building
blocks, called bubbles which are pseudo-manifolds with boundaries. Bubbles can
in turn be glued together to form triangulations. The main challenge is to
classify the triangulations built from a given set of bubbles with respect to
their numbers of bubbles and simplices of codimension two. While the colored
triangulations which maximize the number of simplices of codimension two at
fixed number of simplices are series-parallel objects called melonic
triangulations, this is not always true anymore when restricting attention to
colored triangulations built from specific bubbles. This opens up the
possibility of new universality classes of colored triangulations. We present
three existing strategies to find those universality classes. The first two
strategies consist in building new bubbles from old ones for which the problem
can be solved. The third strategy is a bijection between those colored
triangulations and stuffed, edge-colored maps, which are some sort of hypermaps
whose hyperedges are replaced with edge-colored maps. We then show that the
present approach can lead to enumeration results and identification of
universality classes, by working out the example of quartic tensor models. They
feature a tree-like phase, a planar phase similar to two-dimensional quantum
gravity and a phase transition between them which is interpreted as a
proliferation of baby universes
Tensor network method for reversible classical computation
We develop a tensor network technique that can solve universal reversible classical computational problems, formulated as vertex models on a square lattice [Nat. Commun. 8, 15303 (2017)]. By encoding the truth table of each vertex constraint in a tensor, the total number of solutions compatible with partial inputs and outputs at the boundary can be represented as the full contraction of a tensor network. We introduce an iterative compression-decimation (ICD) scheme that performs this contraction efficiently. The ICD algorithm first propagates local constraints to longer ranges via repeated contraction-decomposition sweeps over all lattice bonds, thus achieving compression on a given length scale. It then decimates the lattice via coarse-graining tensor contractions. Repeated iterations of these two steps gradually collapse the tensor network and ultimately yield the exact tensor trace for large systems, without the need for manual control of tensor dimensions. Our protocol allows us to obtain the exact number of solutions for computations where a naive enumeration would take astronomically long times.We thank Justin Reyes, Oskar Pfeffer, and Lei Zhang for many useful discussions. The computations were carried out at Boston University's Shared Computing Cluster. We acknowledge the Condensed Matter Theory Visitors Program at Boston University for support. Z.-C. Y. and C. C. are supported by DOE Grant No. DE-FG02-06ER46316. E.R.M. is supported by NSF Grant No. CCF-1525943. (Condensed Matter Theory Visitors Program at Boston University; DE-FG02-06ER46316 - DOE; CCF-1525943 - NSF)Accepted manuscrip
Tensorizing Neural Networks
Deep neural networks currently demonstrate state-of-the-art performance in
several domains. At the same time, models of this class are very demanding in
terms of computational resources. In particular, a large amount of memory is
required by commonly used fully-connected layers, making it hard to use the
models on low-end devices and stopping the further increase of the model size.
In this paper we convert the dense weight matrices of the fully-connected
layers to the Tensor Train format such that the number of parameters is reduced
by a huge factor and at the same time the expressive power of the layer is
preserved. In particular, for the Very Deep VGG networks we report the
compression factor of the dense weight matrix of a fully-connected layer up to
200000 times leading to the compression factor of the whole network up to 7
times
Enhancing non-melonic triangulations: A tensor model mixing melonic and planar maps
Ordinary tensor models of rank are dominated at large by
tree-like graphs, known as melonic triangulations. We here show that
non-melonic contributions can be enhanced consistently, leading to different
types of large limits. We first study the most generic quartic model at
, with maximally enhanced non-melonic interactions. The existence of the
expansion is proved and we further characterize the dominant
triangulations. This combinatorial analysis is then used to define a
non-quartic, non-melonic class of models for which the large free energy
and the relevant expectations can be calculated explicitly. They are matched
with random matrix models which contain multi-trace invariants in their
potentials: they possess a branched polymer phase and a 2D quantum gravity
phase, and a transition between them whose entropy exponent is positive.
Finally, a non-perturbative analysis of the generic quartic model is performed,
which proves analyticity in the coupling constants in cardioid domains
Enumeration of uni-singular algebraic hypersurfaces
We enumerate complex algebraic hypersurfaces in , of a given (high)
degree with one singular point of a given singularity type. Our approach is to
compute the (co)homology classes of the corresponding equi-singular strata in
the parameter space of hypersurfaces. We suggest an inductive procedure, based
on intersection theory combined with liftings and degenerations. The procedure
computes the (co)homology class in question, whenever a given singularity type
is properly defined and the stratum possesses good geometric properties. We
consider in details the generalized Newton-non-degenerate singularities. We
give also examples of enumeration in some other cases.Comment: Published versio
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