Markov Logic Networks with Complex Weights and Algorithms to Train Them

Tato práce se zabývá markovskými logickými sítěmi s komplexními vahami (komplexními markovskými logickými sítěmi, C–MLNs). Ty jsou rozšířením markovských logických sítí, které je plně expresivní v kontextu počtových distribucí. Text identifikuje několik problémů s původní C–MLN definicí a navrhuje její úpravu. Dále je odvozena procedura pro inferenci založená na Gibbsově vzorkování. Kromě toho jsou odvozeny dva algoritmy pro učení parametrů. První z nich je založen na metodě maximální věrohodnosti. Z důvodu nekonvexity problému nepodává maximalizace založená na gradietním sestupu dobré výsledky. Druhý algoritmus využívá diskrétní Fourierovu transformaci k vyjádření libovolné počtové distribuce jako C–MLN. Algoritmus nicméně vyžaduje velkou trénovací množinu a učí se váhy o zbytečně vysoké dimenzi.This thesis studies Markov logic networks with complex weights (complex Markov logic networks, C–MLNs). Those are an extension of Markov logic networks that achieves full expressivity in terms of count distributions. Slight modification of the C–MLN definition is proposed attempting to solve a few identified problems. An inference procedure based on Gibbs sampling is developed for the model. Two parameter learning algorithms are proposed as well. The first one utilizes the maximum likelihood estimation. Due to the non-convexity of the problem, gradient descent-based maximization does not perform well. The other learning procedure uses the discrete Fourier transform to encode an arbitrary count distribution as a C–MLN. However, it requires a huge data set, and it learns weights of unnecessarily large dimensions

Lifted Inference with Linear Order Axiom

We consider the task of weighted first-order model counting (WFOMC) used for probabilistic inference in the area of statistical relational learning. Given a formula $\phi$, domain size $n$ and a pair of weight functions, what is the weighted sum of all models of $\phi$ over a domain of size $n$? It was shown that computing WFOMC of any logical sentence with at most two logical variables can be done in time polynomial in $n$. However, it was also shown that the task is \texttt{#}P_1-complete once we add the third variable, which inspired the search for extensions of the two-variable fragment that would still permit a running time polynomial in $n$. One of such extension is the two-variable fragment with counting quantifiers. In this paper, we prove that adding a linear order axiom (which forces one of the predicates in $\phi$ to introduce a linear ordering of the domain elements in each model of $\phi$) on top of the counting quantifiers still permits a computation time polynomial in the domain size. We present a new dynamic programming-based algorithm which can compute WFOMC with linear order in time polynomial in $n$, thus proving our primary claim

Saturated simple and k-simple topological graphs

A simple topological graph $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called saturated if no further edge can be added without violating this condition. We construct saturated simple topological graphs with $n$ vertices and $O(n)$ edges. For every $k>1$, we give similar constructions for $k$-simple topological graphs, that is, for graphs drawn in the plane so that any two edges have at most $k$ points in common. We show that in any $k$-simple topological graph, any two independent vertices can be connected by a curve that crosses each of the original edges at most $2k$ times. Another construction shows that the bound $2k$ cannot be improved. Several other related problems are also considered.Comment: 25 pages, 17 figures, added some new results and improvement

Frontal midline theta connectivity is related to efficiency of WM maintenance and is affected by aging

Representations in working memory (WM) are temporary, but can be refreshed for longer periods of time through maintenance mechanisms, thereby establishing their availability for subsequent memory tests. Frontal brain regions supporting WM maintenance operations undergo anatomical and functional changes with advancing age, leading to age related decline of memory functions. The present study focused on age-related functional connectivity changes of the frontal midline (FM) cortex in the theta band (4–8 Hz), related to WM maintenance. In the visual delayed-match-to-sample WM task young (18–26 years, N = 20) and elderly (60–71 years N = 16) adults had to memorize sample stimuli consisting of 3 or 5 items while 33 channel EEG recording was performed. The phase lag index was used to quantify connectivity strength between cortical regions. The low and high memory demanding WM maintenance periods were classified based on whether they were successfully maintained (remembered) or unsuccessfully maintained (unrecognized later). In the elderly reduced connectivity strength of FM brain region and decreased performance were observed. The connectivity strength between FM and posterior sensory cortices was shown to be sensitive to both increased memory demands and memory performance regardless of age. The coupling of frontal regions (midline and lateral) and FM-temporal cortices characterized successfully maintained trials and declined with advancing age. The findings provide evidence that a FM neural circuit of theta oscillations that serves a possible basis of active maintenance process is especially vulnerable to aging

Long alternating paths in bicolored point sets

AbstractGiven n red and n blue points in convex position in the plane, we show that there exists a noncrossing alternating path of length n+cn/logn. We disprove a conjecture of Erdős by constructing an example without any such path of length greater than 4/3n+c′n