arXiv subjects
Publications and source records attributed to Simon Baker.
In this paper we prove that if $\{φ_i(x)=λx+t_i\}$ is an equicontractive iterated function system and $b$ is a positive integer satisfying $\frac{\log b}{\log |λ|}\notin\mathbb{Q},$ then almost every $x$ is normal in base $b$ for any non-atomic self-similar measure of $\{φ_i\}$.
Let $\lbrace f_i(x)=s_i \cdot x+t_i \rbrace$ be a self-similar IFS on $\mathbb{R}$ and let $β>1$ be a Pisot number. We prove that if $\frac{\log |s_i|}{\log β}\notin \mathbb{Q}$ for some $i$ then for every $C^1$ diffeomorphism $g$ and every non-atomic self similar measure $μ$, the measure $gμ$ is supported on numbers that are normal in base $β$.
We study the learning of a matching model for dialogue response selection. Motivated by the recent finding that models trained with random negative samples are not ideal in real-world scenarios, we propose a hierarchical curriculum learning framework that trains the matching model in an "easy-to-difficult" scheme. Our learning framework consists of two complementary curricula: (1) corpus-level curriculum (CC); and (2) instance-level curriculum (IC). In CC, the model gradually increases its ability in finding the matching clues between the dialogue context and a response candidate. As for IC, it progressively strengthens the model's ability in identifying the mismatching information between the dialogue context and a response candidate. Empirical studies on three benchmark datasets with three state-of-the-art matching models demonstrate that the proposed learning framework significantly improves the model performance across various evaluation metrics.
Neural table-to-text generation models have achieved remarkable progress on an array of tasks. However, due to the data-hungry nature of neural models, their performances strongly rely on large-scale training examples, limiting their applicability in real-world applications. To address this, we propose a new framework: Prototype-to-Generate (P2G), for table-to-text generation under the few-shot scenario. The proposed framework utilizes the retrieved prototypes, which are jointly selected by an IR system and a novel prototype selector to help the model bridging the structural gap between tables and texts. Experimental results on three benchmark datasets with three state-of-the-art models demonstrate that the proposed framework significantly improves the model performance across various evaluation metrics.
Let $M$ be a positive integer and $q\in (1, M+1]$. A $q$-expansion of a real number $x$ is a sequence $(c_i)=c_1c_2\cdots$ with $c_i\in \{0,1,\ldots, M\}$ such that $x=\sum_{i=1}^{\infty}c_iq^{-i}$. In this paper we study the set $\mathcal{U}_q^j$ consisting of those real numbers having exactly $j$ $q$-expansions. Our main result is that for Lebesgue almost every $q\in (q_{KL}, M+1), $ we have $$\dim_{H}\mathcal{U}_{q}^{j}\leq \max\{0, 2\dim_H\mathcal{U}_q-1\}\text{ for all } j\in\{2,3,\ldots\}.$$ Here $q_{KL}$ is the Komornik-Loreti constant. As a corollary of this result, we show that for any $j\in\{2,3,\ldots\},$ the function mapping $q$ to $\dim_{H}\mathcal{U}_{q}^{j}$ is not continuous.
In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchine's theorem for these sets. We then apply this result to the topic of intrinsic Diophantine Approximation on self-similar sets. In particular, we define a new height function for an element of $\mathbb{Q}^d$ contained in a self-similar set in terms of its eventually periodic representations. For limsup sets defined with respect to this height function, we obtain a detailed description of their metric properties. The results of this paper hold in arbitrary dimensions and without any separation conditions on the underlying iterated function system.
A well known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In this paper we give sufficient conditions for an analogue of this theorem to hold for self-similar measures. Our approach applies more generally to sequences of the form $(f_{n}(x))_{n=1}^{\infty}$ where $(f_n)_{n=1}^{\infty}$ is a sequence of sufficiently smooth real valued functions satisfying a nonlinearity assumption. As a corollary of our main result, we show that if $C$ is equal to the middle third Cantor set and $t\geq 1$, then with respect to the Cantor-Lebesgue measure on $C+t$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed for almost every $x$.
Non-autoregressive generation (NAG) has recently attracted great attention due to its fast inference speed. However, the generation quality of existing NAG models still lags behind their autoregressive counterparts. In this work, we show that BERT can be employed as the backbone of a NAG model to greatly improve performance. Additionally, we devise mechanisms to alleviate the two common problems of vanilla NAG models: the inflexibility of prefixed output length and the conditional independence of individual token predictions. Lastly, to further increase the speed advantage of the proposed model, we propose a new decoding strategy, ratio-first, for applications where the output lengths can be approximately estimated beforehand. For a comprehensive evaluation, we test the proposed model on three text generation tasks, including text summarization, sentence compression and machine translation. Experimental results show that our model significantly outperforms existing non-autoregressive baselines and achieves competitive performance with many strong autoregressive models. In addition, we also conduct extensive analysis experiments to reveal the effect of each proposed component.
In this paper we obtain new lower bounds for the upper box dimension of $αβ$ sets. As a corollary of our main result, we show that if $α$ is not a Liouville number and $β$ is a Liouville number, then the upper box dimension of any $αβ$ set is $1$. We also use our dimension bounds to obtain new results on affine embeddings of self-similar sets.
Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $α>1$, the fractional parts of the geometric progression $(α^{n})_{n\geq1}$ are equidistributed modulo one. In the present paper we sharpen this result by showing that for almost every $α>1$, the correlations of all finite orders and hence the normalized gaps of $(α^{n})_{n\geq1}$ mod 1 have a Poissonian limit distribution, thereby resolving a conjecture of the two first named authors. While an earlier approach used probabilistic methods in the form of martingale approximation, our reasoning in the present paper is of an analytic nature and based upon the estimation of oscillatory integrals. This method is robust enough to allow us to extend our results to a natural class of sub-lacunary sequences.
In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation. This result shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems our results apply to include those arising from Bernoulli convolutions, the $\{0,1,3\}$ problem, and affine contractions with varying translation parameter. As a by-product of our analysis we obtain new proofs of well known results due to Solomyak on the absolute continuity of Bernoulli convolutions, and when the attractor in the $\{0,1,3\}$ problem has positive Lebesgue measure. For each $t\in [0,1]$ we let $Φ_t$ be the iterated function system given by $$Φ_{t}:=\Big\{ϕ_1(x)=\frac{x}{2},ϕ_2(x)=\frac{x+1}{2},ϕ_3(x)=\frac{x+t}{2},ϕ_{4}(x)=\frac{x+1+t}{2}\Big\}.$$ We include a detailed study of this family. We prove that either $Φ_t$ contains an exact overlap, or we observe Khintchine like behaviour. Our analysis of this family shows that by studying the metric properties of limsup sets, we can distinguish between the overlapping behaviour of iterated function systems in a way that is not available to us by simply studying properties of self-similar measures. Last of all, we introduce a property of an iterated function system that we call being consistently separated with respect to a measure. We prove that this property implies that the pushforward of the measure is absolutely continuous. We include several explicit examples of consistently separated iterated function systems.
In this paper we study random iterated function systems. Our main result gives sufficient conditions for an analogue of a well known theorem due to Khintchine from Diophantine approximation to hold almost surely for stochastically self-similar and self-affine random iterated function systems.
In this paper we study the quantitative recurrence properties of self-conformal sets $X$ equipped with the map $T:X\to X$ induced by the left shift. In particular, given a function $φ:\mathbb{N}\to(0,\infty),$ we study the metric properties of the set $$R(T,φ)=\left\{x\in X:|T^nx-x|<φ(n)\textrm{ for infinitely many }n\in \mathbb{N}\right\}.$$ Our main result shows that for the natural measure supported on $X$, $R(T,φ)$ has zero measure if a natural volume sum converges, and under the open set condition $R(T,φ)$ has full measure if this volume sum diverges.
Until recently, it was an important open problem in Fractal Geometry to determine whether there exists an iterated function system acting on $\mathbb{R}$ with no exact overlaps for which cylinders are super-exponentially close at all small scales. Iterated function systems satisfying these properties were shown to exist by the author and by Bárány and Käenmäki. In this paper we prove a general theorem on the existence of such iterated function systems within a parameterised family. This theorem shows that if a parameterised family contains two independent subfamilies, and the set of parameters that cause exact overlaps satisfies some weak topological assumptions, then the original family will contain an iterated function system satisfying the desired properties. We include several explicit examples of parameterised families to which this theorem can be applied.
Khintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFSs). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and satisfies the open set condition. The divergence condition we introduce incorporates the inhomogeneity present within the IFS. We demonstrate via an example that such an approach is essential. We also formulate an analogue of the Duffin-Schaeffer conjecture and show that it holds for a set of full Hausdorff dimension. Combining our results with the mass transference principle of Beresnevich and Velani \cite{BerVel}, we prove a general result that implies the existence of exceptional points within the attractor of our IFS. These points are exceptional in the sense that they are "very well approximated". As a corollary of this result, we obtain a general solution to a problem of Mahler, and prove that there are badly approximable numbers that are very well approximated by quadratic irrationals. The ideas put forward in this paper are introduced in the general setting of IFSs that may contain overlaps. We believe that by viewing IFS's from the perspective of metric number theory, one can gain a greater insight into the extent to which they overlap. The results of this paper should be interpreted as a first step in this investigation.
Stylistic response generation is crucial for building an engaging dialogue system for industrial use. While it has attracted much research interest, existing methods often generate stylistic responses at the cost of the content quality (relevance and fluency). To enable better balance between the content quality and the style, we introduce a new training strategy, know as Information-Guided Reinforcement Learning (IG-RL). In IG-RL, a training model is encouraged to explore stylistic expressions while being constrained to maintain its content quality. This is achieved by adopting reinforcement learning strategy with statistical style information guidance for quality-preserving explorations. Experiments on two datasets show that the proposed approach outperforms several strong baselines in terms of the overall response performance.
The ability of a dialog system to express prespecified language style during conversations has a direct, positive impact on its usability and on user satisfaction. We introduce a new prototype-to-style (PS) framework to tackle the challenge of stylistic dialogue generation. The framework uses an Information Retrieval (IR) system and extracts a response prototype from the retrieved response. A stylistic response generator then takes the prototype and the desired language style as model input to obtain a high-quality and stylistic response. To effectively train the proposed model, we propose a new style-aware learning objective as well as a de-noising learning strategy. Results on three benchmark datasets from two languages demonstrate that the proposed approach significantly outperforms existing baselines in both in-domain and cross-domain evaluations
We introduce Multi-SimLex, a large-scale lexical resource and evaluation benchmark covering datasets for 12 typologically diverse languages, including major languages (e.g., Mandarin Chinese, Spanish, Russian) as well as less-resourced ones (e.g., Welsh, Kiswahili). Each language dataset is annotated for the lexical relation of semantic similarity and contains 1,888 semantically aligned concept pairs, providing a representative coverage of word classes (nouns, verbs, adjectives, adverbs), frequency ranks, similarity intervals, lexical fields, and concreteness levels. Additionally, owing to the alignment of concepts across languages, we provide a suite of 66 cross-lingual semantic similarity datasets. Due to its extensive size and language coverage, Multi-SimLex provides entirely novel opportunities for experimental evaluation and analysis. On its monolingual and cross-lingual benchmarks, we evaluate and analyze a wide array of recent state-of-the-art monolingual and cross-lingual representation models, including static and contextualized word embeddings (such as fastText, M-BERT and XLM), externally informed lexical representations, as well as fully unsupervised and (weakly) supervised cross-lingual word embeddings. We also present a step-by-step dataset creation protocol for creating consistent, Multi-Simlex-style resources for additional languages. We make these contributions -- the public release of Multi-SimLex datasets, their creation protocol, strong baseline results, and in-depth analyses which can be be helpful in guiding future developments in multilingual lexical semantics and representation learning -- available via a website which will encourage community effort in further expansion of Multi-Simlex to many more languages. Such a large-scale semantic resource could inspire significant further advances in NLP across languages.