In text mining, topic models are a type of probabilistic generative models for inferring latent semantic topics from text corpus. One of the most popular inference approaches to topic models …
In text mining, topic models are a type of probabilistic generative models for inferring latent semantic topics from text corpus. One of the most popular inference approaches to topic models is perhaps collapsed Gibbs sampling (CGS), which typically samples one single topic label for each observed document-word pair. In this paper, we aim at improving the inference of CGS for topic models. We propose to leverage state augmentation technique by maximizing the number of topic samples to infinity, and then develop a new inference approach, called infinite latent state replication (ILR), to generate robust soft topic assignment for each given document-word pair. Experimental results on the publicly available datasets show that ILR outperforms CGS for inference of existing established topic models.
We generalize the notions of definable amenability and extreme definable amenability to continuous structures and show that the stable and ultracompact groups are definable amenable. In addition, we characterize both …
We generalize the notions of definable amenability and extreme definable amenability to continuous structures and show that the stable and ultracompact groups are definable amenable. In addition, we characterize both notions in terms of fixed-point properties. We prove that, for dependent theories, definable amenability is equivalent to the existence of a good S1 ideal. Finally, we show the randomizations of first-order definable amenable groups are extremely definably amenable.
Based on the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of …
Based on the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking-independent elements that is dense inside a partial type $\mathcal{G}(x)$, which we call $H$-structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again $H$-structures. We prove that under these assumptions the expansion is supersimple and characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one-basedness and CM-triviality. In the one-based case, when $T$ has $SU$-rank $\omega^\alpha$ and the $SU$-rank is continuous, we take $\mathcal{G}(x)$ to be the type of elements of $SU$-rank $\omega^\alpha$ and we describe a natural geometry of generics modulo $H$ associated with such expansions and show it is modular.
Based on the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of …
Based on the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking-independent elements that is dense inside a partial type $\mathcal{G}(x)$, which we call $H$-structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again $H$-structures. We prove that under these assumptions the expansion is supersimple and characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one-basedness and CM-triviality. In the one-based case, when $T$ has $SU$-rank $\omega^\alpha$ and the $SU$-rank is continuous, we take $\mathcal{G}(x)$ to be the type of elements of $SU$-rank $\omega^\alpha$ and we describe a natural "geometry of generics modulo $H$" associated with such expansions and show it is modular.
Abstract Based on the work done in [][] in the o‐minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set …
Abstract Based on the work done in [][] in the o‐minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking‐independent elements that is dense inside a partial type , which we call H ‐structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again H ‐structures. We prove that under these assumptions the expansion is supersimple and characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one‐basedness and CM‐triviality. In the one‐based case, when T has SU‐rank and the SU‐rank is continuous, we take to be the type of elements of SU‐rank and we describe a natural “geometry of generics modulo H ” associated with such expansions and show it is modular.
We prove an analogue of the fixed-point theorem for the case of definably amenable groups.
We prove an analogue of the fixed-point theorem for the case of definably amenable groups.
The main result of this thesis is the study of how amplenessgrows in geometric and SU-rank omega structures when adding anew independent dense/codense subset. In another direction, we explorerelations of …
The main result of this thesis is the study of how amplenessgrows in geometric and SU-rank omega structures when adding anew independent dense/codense subset. In another direction, we explorerelations of ampleness with equational theories; there, we give a directproof of the equationality of certain CM-trivial theories. Finally, we studyindiscernible closed sets—which are closely related with equations—andmeasure their complexity in the free pseudoplane.
Extending the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory of SU-rank $\omega$ with a dense codense independent collection$H$ …
Extending the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory of SU-rank $\omega$ with a dense codense independent collection$H$ of element of rank $\omega$, where density of means it intersectsany definable set of $SU$-rank omega. We show that under some technical conditions, the class of such structures is first order.We prove that the expansion is supersimple and characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one-basedness and CM-triviality. In the one-based case, we describe a natural geometry of generics modulo $H$ associated with such expansions and show it is modular.
Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but …
Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but non-isomorphic fundamental groups. To do so, the Alexander Invariant and certain invariants of combinatorial line arrangements are presented and developed for combinatorics with only double and triple points. This is part of a more general project to better understand the relationship between topology and combinatorics of line arrangements.
CM -triviality of a stable theory is a notion introduced by Hrushovski [1]. The importance of this property is first that it holds of Hrushovski's new non 1-based strongly minimal …
CM -triviality of a stable theory is a notion introduced by Hrushovski [1]. The importance of this property is first that it holds of Hrushovski's new non 1-based strongly minimal sets, and second that it is still quite a restrictive property, and forbids the existence of definable fields or simple groups (see [2]). In [5], Frank Wagner posed some questions about CM -triviality, asking in particular whether a structure of finite rank, which is “coordinatized” by CM -trivial types of rank 1, is itself CM -trivial. (Actually Wagner worked in a slightly more general context, adapting the definitions to a certain “local” framework, in which algebraic closure is replaced by P -closure, for P some family of types. We will, however, remain in the standard context, and will just remark here that it is routine to translate our results into Wagner's framework, as well as to generalise to the superstable theory/regular type context.) In any case we answer Wagner's question positively. Also in an attempt to put forward some concrete conjectures about the possible geometries of strongly minimal sets (or stable theories) we tentatively suggest a hierarchy of geometric properties of forking, the first two levels of which correspond to 1-basedness and CM -triviality respectively. We do not know whether this is a strict hierarchy (or even whether these are the “right” notions), but we conjecture that it is, and moreover that a counterexample to Cherlin's conjecture can be found at level three in the hierarchy.
Abstract The notion of CM-triviality was introduced by Hrushovski, who showed that his new strongly minimal sets have this property. Recently Baudisch has shown that his new ω 1 -categorical …
Abstract The notion of CM-triviality was introduced by Hrushovski, who showed that his new strongly minimal sets have this property. Recently Baudisch has shown that his new ω 1 -categorical group has this property. Here we show that any group of finite Morley rank definable in a CM-trivial theory is nilpotent-by-finite, or equivalently no simple group of finite Morley rank can be definable in a CM-trivial theory.
Abstract We give a uniform construction of free pseudospaces of dimension n extending work in [1]. This yields examples of ω -stable theories which are n -ample, but not n …
Abstract We give a uniform construction of free pseudospaces of dimension n extending work in [1]. This yields examples of ω -stable theories which are n -ample, but not n + 1-ample. The prime models of these theories are buildings associated to certain right-angled Coxeter groups.
Le paradigme de théorie stable est la théorie T d'un corps algébriquement clos; une autre théorie stable T ′ est celle de la structure formée d'un corps algébriquement clos, avec …
Le paradigme de théorie stable est la théorie T d'un corps algébriquement clos; une autre théorie stable T ′ est celle de la structure formée d'un corps algébriquement clos, avec en outre un symbole relationnel unaire interprétant un de ses sous-corps propres algébriquement clos. C'est à l'éclaircissement des rapports de T et de T ′ qu'est consacré cet article. J'y considère une théorie complète T stable, et les structures formées d'un modèle N de T, avec en outre un symbole relationnel unaire ( x ) interprétant une restriction élémentaire M de N ; j'appelle ces structures paires de modèles de T . Et je dis que la paire ( N, M ) est belle si d'une part M est ∣ T ∣ + -saturé, et d'autre part pour tout n -uplet ā d'éléments de N , tout type, au sens de T , sur M ⋃ {α} est réalisé dans N . Le premier résultat (Théorème 4) est que deux belles paires sont élémentairement équivalentes. Plus précisément, si ( N 1 , M 1 ) et ( N 2 , M 2 ) sont deux belles paires, et si ā est dans la première, b¯ dans la seconde, le fait que le type de ā sur M 1 et celui de b¯ sur M 2 soient équivalents dans l'ordre fondamental au sens de T suffit (et est bien sûr nécessaire) pour que ā at b¯ aient même type (sur ⊘) au sens de la théorie T ′ des belles paires.
Let M be an L -structure and A be an infinite subset of M . Two structures can be defined from A : • The induced structure on A has …
Let M be an L -structure and A be an infinite subset of M . Two structures can be defined from A : • The induced structure on A has a name R φ for every ∅-definable relation φ ( M ) ∩ A n on A . Its language is A with its L ind -structure will be denoted by A ind . • The pair ( M, A ) is an L(P) -structure, where P is a unary predicate for A and L(P) = L ∪{ P }. We call A small if there is a pair ( N, B ) elementarily equivalent to ( M, A ) and such that for every finite subset b of N every L –type over Bb is realized in N . A formula φ ( x, y ) has the finite cover property (f.c.p) in M if for all natural numbers k there is a set of φ –formulas which is k –consistent but not consistent in M. M has the f.c.p if some formula has the f.c.p in M . It is well known that unstable structures have the f.c.p. (see [6].) We will prove the following two theorems. Theorem A. Let A be a small subset of M. If M does not have the finite cover property then, for every λ ≥ ∣ L ∣, if both M and A ind are λ– stable then (M, A) is λ– stable . Corollary 1.1 (Poizat [5]). If M does not have the finite cover property and N ≺ M is a small elementary substructure, then (M, N) is stable . Corollary 1.2 (Zilber [7]). If U is the group of wots of unity in the field ℂ of complex numbers the pair (ℂ, U ) is ω – stable . Proof. (See [4].) As a strongly minimal set ℂ is ω–stable and does not have the f.c.p. By the subspace theorem of Schmidt [3] every algebraic set intersects U in a finite union of translates of subgroups definable in the group structure of U alone. Whence U ind is nothing more than a (divisible) abelian group, which is ω –stable.
At the source of what is now known as “geometric stability theory” was Zil'ber's intuition that the essential properties of an aleph-one-categorical theory were controlled by the geometries of its …
At the source of what is now known as “geometric stability theory” was Zil'ber's intuition that the essential properties of an aleph-one-categorical theory were controlled by the geometries of its minimal types. (However, the situation is much more complex than was assumed in Zil'ber [1984], since the main conjecture of that paper has been disproved by Hrushovski.) This is not unnatural in this unidimensional case, where all these geometries have isomorphic contractions, but it was even realized later, in Cherlin, Harrington and Lachlan [1985] and Buechler [1986], that, for any superstable theory with finite ranks, a certain “local” property, i.e. a property satisfied by the geometry of each type of rank one (namely: to have a projective contraction), was equivalent to a “global” one (the theory is one-based, hence satisfies a coordinatization lemma). Then it was shown, in Pillay [1986], that this situation does not generalize to the infinite rank case, that, even for a theory of rank omega, the (local) assumption of projectivity for all the regular types of the theory does not have an exact global counterpart. To clarify this kind of phenomena, I suggest here the elimination of their geometrical aspect, considering only the case where all of the geometries are degenerate. I will study various notions of triviality, which make sense in a stable context, and turn out to be equivalent in the finite rank case; some of them have a definite global flavour, others are of local character.
In this paper we construct a non- CM -trivial stable theory in which no infinite field is interpretable. In fact our theory will also be trivial and ω-stable, but of …
In this paper we construct a non- CM -trivial stable theory in which no infinite field is interpretable. In fact our theory will also be trivial and ω-stable, but of infinite Morley rank. A long term aim would be to find a non CM -trivial theory which has finite Morley rank (or is even strongly minimal) and does not interpret a field. The construction in this paper is direct, and is a “3-dimensional” version of the free pseudoplane. In a sense we are cheating: the original point of the notion of CM -triviality was to describe the geometry of a strongly minimal set, or even of a regular type. In our example, non- CM -triviality will come from the behaviour of three orthogonal regular types. A stable theory is said to be CM -trivial if whenever A ⊆ B and acl( Ac ) ∩ acl( B ) = acl( A ) in T eq , then Cb(stp( c / A )) ⊆ Cb(stp( c / B )). ( An infinite stable field will not be CM -trivial.) The notion is due to Hrushovski [3], where he gave several equivalent definitions, as well as showing that his new strongly minimal sets constructed “ab ovo” were CM -trivial. The notion was studied further in [6] where it was shown that CM -trivial groups of finite Morley rank are nilpotent-by-finite. These results were generalized in various ways to the superstable case in [8].
We use pseudo-Anosov homeomorphisms of surfaces in order to prove that the first-order theory of non-Abelian free groups, T_{fg}, is n-ample for any n \in \omega. This result adds to …
We use pseudo-Anosov homeomorphisms of surfaces in order to prove that the first-order theory of non-Abelian free groups, T_{fg}, is n-ample for any n \in \omega. This result adds to the work of Pillay, which proved that T_{fg} is non-CM-trivial. The sequence witnessing ampleness is a sequence of primitive elements in F_{\omega}. Our result provides an alternative proof to the main result of a recent preprint by Ould Houcine and Tent.
Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find …
Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω -categorical geometric theory interprets an infinite vector space over a finite field.
An impressive theory has been developed, largely by Shelah, around the notion of a stable theory. This includes detailed structure theorems for the models of such theories as well as …
An impressive theory has been developed, largely by Shelah, around the notion of a stable theory. This includes detailed structure theorems for the models of such theories as well as a generalized notion of independence. The various stability properties can be defined in terms of the numbers of types over sets, or in terms of the complexity of definable sets. In the concrete examples of stable theories, however, one finds an important distinction between “positive” and “negative” information, such a distinction not being an a priori consequence of the general definitions. In the naive examples this may take the form of distinguishing between say a class of a definable equivalence relation and the complement of a class. In the more algebraic examples, this distinction may have a “topological” significance, for example with the Zariski topology on (the set of n -tuples of) an algebraically closed field, the “closed” sets being those given by sets of polynomial equalities. Note that in the latter case, every definable set is a Boolean combination of such closed sets (the definable sets are precisely the constructible sets). Similarly, stability conditions in practice reduce to chain conditions on certain “special” definable sets (e.g. in modules, stable groups). The aim here is to develop and present such notions in the general (model-theoretic) context. The basic notion is that of an “equation”. Given a complete theory T in a language L , an L -formula φ ( x̄, ȳ ) is said to be an equation (in x̄ ) if any collection Φ of instances of φ (i.e. of formulae φ ( x̄, ā )) is equivalent to a finite subset Φ′ ⊂ Φ .
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating …
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.
We associate with every first order structure [Formula: see text] a family of invariant, locally Noetherian topologies (one topology on each M n ). The structure is almost determined by …
We associate with every first order structure [Formula: see text] a family of invariant, locally Noetherian topologies (one topology on each M n ). The structure is almost determined by the topologies, and properties of the structure are reflected by topological properties. We study these topologies in particular for stable structures. In nice cases, we get a behaviour similar to the Zariski topology in algebraically closed fields.
We show that if T is any geometric theory having the NTP 2 then the corresponding theories of lovely pairs of models of T and of H ‐structures associated to …
We show that if T is any geometric theory having the NTP 2 then the corresponding theories of lovely pairs of models of T and of H ‐structures associated to T also have the NTP 2 . We also prove that if T is strong then the same two expansions of T are also strong.
We study the action of $G = {\rm SL} (2,\mathbb R)$, viewed as a group definable in the structure $M = (\mathbb R,+,\times )$, on its type space $S_{G}(M)$. We …
We study the action of $G = {\rm SL} (2,\mathbb R)$, viewed as a group definable in the structure $M = (\mathbb R,+,\times )$, on its type space $S_{G}(M)$. We identify a minimal closed $G$-flow $I$ and an idempotent $r\in I$ (with respect to the Ellis se
In a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries. Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial …
In a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries. Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial theory, the independence relation is stable.
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine …
Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.
We show that the theory of the free group – and more generally the theory of any torsionfree hyperbolic group – is n-ample for any n ≥ 1. We give …
We show that the theory of the free group – and more generally the theory of any torsionfree hyperbolic group – is n-ample for any n ≥ 1. We give also an explicit description of the imaginary algebraic closure in free groups.
Abstract We construct a stable one-based, trivial theory with a reduct which is not trivial. This answers a question of John B. Goode. Using this, we construct a stable theory …
Abstract We construct a stable one-based, trivial theory with a reduct which is not trivial. This answers a question of John B. Goode. Using this, we construct a stable theory which is n -ample for all natural numbers n , and does not interpret an infinite group.
We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, …
We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.
We give an example of a definable set in every free or torsion-free (non-elementary) hyperbolic group that is not in the Boolean algebra of equational sets. Hence, the theories of …
We give an example of a definable set in every free or torsion-free (non-elementary) hyperbolic group that is not in the Boolean algebra of equational sets. Hence, the theories of free and torsion-free (non-elementary) hyperbolic groups are not equational in the sense of G. Srour.
Let X be a Zariski open subset of a compact Kaehler manifold. In this paper, we study the set $Σ^k(X)$ of one dimensional local systems on X with nonvanishing kth …
Let X be a Zariski open subset of a compact Kaehler manifold. In this paper, we study the set $Σ^k(X)$ of one dimensional local systems on X with nonvanishing kth cohomology. We show that under certain conditions (X compact, X has a smooth compactification with trivial first Betti number, or k=1) $Σ^k(X)$ is a union of translates of sets of the form $f^*H^1(T,C^*)$, where $f:X \to T$ is a holomorphic map to a complex Lie group which is an extension of a compact complex torus by a product of C^*'s (these correspond to semiabelian varieties in the algebraic category). This generalizes earlier work of Beauville, Green, Lazarsfeld, Simpson and the author in the compact case. The main novelty lies in the proofs which involve consideration of Higgs fields with logarithmic poles. While a completely satifactory theory of such objects is still lacking, we are able to work out what we need in the rank one case by borrowing ideas from mixed Hodge theory. This will appear in the Journal of Algebraic Geometry.
An axiomatic treatment of `independence relations' (notions of independence) for complete first-order theories is presented, the principal examples being forking (due to Shelah) and thorn-forking (due to Onshuus). Thorn-forking is …
An axiomatic treatment of `independence relations' (notions of independence) for complete first-order theories is presented, the principal examples being forking (due to Shelah) and thorn-forking (due to Onshuus). Thorn-forking is characterised in terms of modular pairs in the lattice of algebraically closed sets. Wherever possible, forking and thorn-forking are treated in a uniform way. They are dual in the sense that forking is the finest (most restrictive) and thorn-forking the coarsest independence relation worth examining. We finish by defining the kernel of a sequence of indiscernibles and studying its relation to canonical bases.
We construct a new non-desarguesian projective plane from a complex analytic structure. At the same time the construction can be explained in terms of so called Hrushovski's construction. This supports …
We construct a new non-desarguesian projective plane from a complex analytic structure. At the same time the construction can be explained in terms of so called Hrushovski's construction. This supports the hypothesis that in general structures produced by Hrushovski's construction have "prototypes" in complex geometry.
fixed point property. This is the name given by Furstenberg to groups which have a fixed point every time they act affinely on a compact convex set in a locally …
fixed point property. This is the name given by Furstenberg to groups which have a fixed point every time they act affinely on a compact convex set in a locally convex topological linear space. Day [3] has shown that amenability implies the fixed point property. For discrete groups he has shown the converse. Along with amenable groups, we shall study, in this paper, groups with the fixed point property. This paper is based on part of the author's Ph.D. dissertation at Yale University. The author wishes to express his thanks to his adviser, F. J. Hahn. NOTATION. Group will always mean topological group. For a group G, Go will denote the identity component. Likewise Ho will be the identity component of H, etc. Banach spaces, topological vector spaces, etc., will always be over the real field. For topology, we use the notation of Kelley [18], except that our spaces will always