46,326 research outputs found
Multi-Operational Mathematical Derivations in Latent Space
This paper investigates the possibility of approximating multiple
mathematical operations in latent space for expression derivation. To this end,
we introduce different multi-operational representation paradigms, modelling
mathematical operations as explicit geometric transformations. By leveraging a
symbolic engine, we construct a large-scale dataset comprising 1.7M derivation
steps stemming from 61K premises and 6 operators, analysing the properties of
each paradigm when instantiated with state-of-the-art neural encoders.
Specifically, we investigate how different encoding mechanisms can approximate
equational reasoning in latent space, exploring the trade-off between learning
different operators and specialising within single operations, as well as the
ability to support multi-step derivations and out-of-distribution
generalisation. Our empirical analysis reveals that the multi-operational
paradigm is crucial for disentangling different operators, while discriminating
the conclusions for a single operation is achievable in the original expression
encoder. Moreover, we show that architectural choices can heavily affect the
training dynamics, structural organisation, and generalisation of the latent
space, resulting in significant variations across paradigms and classes of
encoders
A Tutorial on Bayesian Nonparametric Models
A key problem in statistical modeling is model selection, how to choose a
model at an appropriate level of complexity. This problem appears in many
settings, most prominently in choosing the number ofclusters in mixture models
or the number of factors in factor analysis. In this tutorial we describe
Bayesian nonparametric methods, a class of methods that side-steps this issue
by allowing the data to determine the complexity of the model. This tutorial is
a high-level introduction to Bayesian nonparametric methods and contains
several examples of their application.Comment: 28 pages, 8 figure
Why do spatial abilities predict mathematical performance?
Spatial ability predicts performance in mathematics and eventual expertise in science, technology and engineering. Spatial skills have also been shown to rely on neuronal networks partially shared with mathematics. Understanding the nature of this association can inform educational practices and intervention for mathematical underperformance. Using data on two aspects of spatial ability and three domains of mathematical ability from 4174 pairs of 12-year-old twins, we examined the relative genetic and environmental contributions to variation in spatial ability and to its relationship with different aspects of mathematics. Environmental effects explained most of the variation in spatial ability (~70%) and in mathematical ability (~60%) at this age, and the effects were the same for boys and girls. Genetic factors explained about 60% of the observed relationship between spatial ability and mathematics, with a substantial portion of the relationship explained by common environmental influences (26% and 14% by shared and non-shared environments respectively). These findings call for further research aimed at identifying specific environmental mediators of the spatial–mathematics relationship
Quantum Interaction Approach in Cognition, Artificial Intelligence and Robotics
The mathematical formalism of quantum mechanics has been successfully
employed in the last years to model situations in which the use of classical
structures gives rise to problematical situations, and where typically quantum
effects, such as 'contextuality' and 'entanglement', have been recognized. This
'Quantum Interaction Approach' is briefly reviewed in this paper focusing, in
particular, on the quantum models that have been elaborated to describe how
concepts combine in cognitive science, and on the ensuing identification of a
quantum structure in human thought. We point out that these results provide
interesting insights toward the development of a unified theory for meaning and
knowledge formalization and representation. Then, we analyze the technological
aspects and implications of our approach, and a particular attention is devoted
to the connections with symbolic artificial intelligence, quantum computation
and robotics.Comment: 10 page
Quantum Theory and Conceptuality: Matter, Stories, Semantics and Space-Time
We elaborate the new interpretation of quantum theory that we recently
proposed, according to which quantum particles are considered conceptual
entities mediating between pieces of ordinary matter which are considered to
act as memory structures for them. Our aim is to identify what is the
equivalent for the human cognitive realm of what physical space-time is for the
realm of quantum particles and ordinary matter. For this purpose, we identify
the notion of 'story' as the equivalent within the human cognitive realm of
what ordinary matter is in the physical quantum realm, and analyze the role
played by the logical connectives of disjunction and conjunction with respect
to the notion of locality. Similarly to what we have done in earlier
investigations on this new quantum interpretation, we use the specific
cognitive environment of the World-Wide Web to elucidate the comparisons we
make between the human cognitive realm and the physical quantum realm.Comment: 14 page
A Potentiality and Conceptuality Interpretation of Quantum Physics
We elaborate on a new interpretation of quantum mechanics which we introduced
recently. The main hypothesis of this new interpretation is that quantum
particles are entities interacting with matter conceptually, which means that
pieces of matter function as interfaces for the conceptual content carried by
the quantum particles. We explain how our interpretation was inspired by our
earlier analysis of non-locality as non-spatiality and a specific
interpretation of quantum potentiality, which we illustrate by means of the
example of two interconnected vessels of water. We show by means of this
example that philosophical realism is not in contradiction with the recent
findings with respect to Leggett's inequalities and their violations. We
explain our recent work on using the quantum formalism to model human concepts
and their combinations and how this has given rise to the foundational ideas of
our new quantum interpretation. We analyze the equivalence of meaning in the
realm of human concepts and coherence in the realm of quantum particles, and
how the duality of abstract and concrete leads naturally to a Heisenberg
uncertainty relation. We illustrate the role played by interference and
entanglement and show how the new interpretation explains the problems related
to identity and individuality in quantum mechanics. We put forward a possible
scenario for the emergence of the reality of macroscopic objects.Comment: 20 pages, 1 figur
Recommended from our members
Coordinating visualizations of polysemous action: Values added for grounding proportion
We contribute to research on visualization as an epistemic learning tool by inquiring into the didactical potential of having students visualize one phenomenon in accord with two different partial meanings of the same concept. 22 Grade 4-6 students participated in a design study that investigated the emergence of proportional-equivalence notions from mediated perceptuomotor schemas. Working as individuals or pairs in tutorial clinical interviews, students solved non-symbolic interaction problems that utilized remote-sensing technology. Next, they used symbolic artifacts interpolated into the problem space as semiotic means to objectify in mathematical register a variety of both additive and multiplicative solution strategies. Finally, they reflected on tensions between these competing visualizations of the space. Micro-ethnographic analyses of episodes from three paradigmatic case studies suggest that students reconciled semiotic conflicts by generating heuristic logico-mathematical inferences that integrated competing meanings into cohesive conceptual networks. These inferences hinged on revisualizing additive elements multiplicatively. Implications are drawn for rethinking didactical design for proportions. © 2013 FIZ Karlsruhe
Graphs for margins of Bayesian networks
Directed acyclic graph (DAG) models, also called Bayesian networks, impose
conditional independence constraints on a multivariate probability
distribution, and are widely used in probabilistic reasoning, machine learning
and causal inference. If latent variables are included in such a model, then
the set of possible marginal distributions over the remaining (observed)
variables is generally complex, and not represented by any DAG. Larger classes
of mixed graphical models, which use multiple edge types, have been introduced
to overcome this; however, these classes do not represent all the models which
can arise as margins of DAGs. In this paper we show that this is because
ordinary mixed graphs are fundamentally insufficiently rich to capture the
variety of marginal models.
We introduce a new class of hyper-graphs, called mDAGs, and a latent
projection operation to obtain an mDAG from the margin of a DAG. We show that
each distinct marginal of a DAG model is represented by at least one mDAG, and
provide graphical results towards characterizing when two such marginal models
are the same. Finally we show that mDAGs correctly capture the marginal
structure of causally-interpreted DAGs under interventions on the observed
variables
Searching for an equation: Dirac, Majorana and the others
We review the non-trivial issue of the relativistic description of a quantum
mechanical system that, contrary to a common belief, kept theoreticians busy
from the end of 1920s to (at least) mid 1940s. Starting by the well-known works
by Klein-Gordon and Dirac, we then give an account of the main results achieved
by a variety of different authors, ranging from de Broglie to Proca, Majorana,
Fierz-Pauli, Kemmer, Rarita-Schwinger and many others.
A particular interest comes out for the general problem of the description of
particles with \textit{arbitrary} spin, introduced (and solved) by Majorana as
early as 1932, and later reconsidered, within a different approach, by Dirac in
1936 and by Fierz-Pauli in 1939. The final settlement of the problem in 1945 by
Bhabha, who came back to the general ideas introduced by Majorana in 1932, is
discussed as well, and, by making recourse also to unpublished documents by
Majorana, we are able to reconstruct the line of reasoning behind the Majorana
and the Bhabha equations, as well as its evolution. Intriguingly enough, such
an evolution was \textit{identical} in the two authors, the difference being
just the period of time required for that: probably few weeks in one case
(Majorana), while more than ten years in the other one (Bhabha), with the
contribution of several intermediate authors.
Majorana's paper of 1932, in fact, contrary to the more complicated
Dirac-Fierz-Pauli formalism, resulted to be very difficult to fully understand
(probably for its pregnant meaning and latent physical and mathematical
content): as is clear from his letters, even Pauli (who suggested its reading
to Bhabha) took about one year in 1940-1 to understand it. This just testifies
for the difficulty of the problem, and for the depth of Majorana's reasoning
and results.Comment: amsart, 34 pages, no figure
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