Every second, an enormous volume of data is being gathered from various sources and stored in huge data banks. Most of the time, monitoring a data source requires several parallel measurements, which form a high-dimensional sample vector. Due to the curse of dimensionality, applying machine learning methods, that is, studying and analyzing high-dimensional data, could be difficult. The essential task of dimensionality reduction is to faithfully represent a given set of high-dimensional data samples with a few variables. The goal of this thesis is to develop and propose new techniques for handling high-dimensional data, in order to address contemporary demand in machine learning applications. Most prominent nonlinear dimensionality reduction methods do not explicitly provide a way to handle out-of-samples. The starting point of this thesis is a nonlinear technique, called Embedding by Affine Transformations (EAT), which reduces the dimensionality of out-of-sample data as well. In this method, a convex optimization is solved for estimating a transformation between the high-dimensional input space and the low-dimensional embedding space. To the best of our knowledge, EAT is the only distance-preserving method for nonlinear dimensionality reduction capable of handling out-of-samples. The second method that we propose is TesseraMap. This method is a scalable extension of EAT. Conceptually, TesseraMap partitions the underlying manifold of data into a set of tesserae and then unfolds it by constructing a tessellation in a low-dimensional subspace of the embedding space. Crucially, the desired tessellation is obtained through solving a small semidefinite program; therefore, this method can efficiently handle tens of thousands of data points in a short time. The final outcome of this thesis is a novel method in dimensionality reduction called Isometric Patch Alignment (IPA). Intuitively speaking, IPA first considers a number of overlapping flat patches, which cover the underlying manifold of the high-dimensional input data. Then, IPA rearranges the patches and stitches the neighbors together on their overlapping parts. We prove that stitching two neighboring patches aligns them together; thereby, IPA unfolds the underlying manifold of data. Although this method and TesseraMap have similar approaches, IPA is more scalable; it embeds one million data points in only a few minutes. More importantly, unlike EAT and TesseraMap, which unfold the underlying manifold by stretching it, IPA constructs the unfolded manifold through patch alignment. We show this novel approach is advantageous in many cases. In addition, compared to the other well-known dimensionality reduction methods, IPA has several important characteristics; for example, it is noise tolerant, it handles non-uniform samples, and it can embed non-convex manifolds properly. In addition to these three dimensionality reduction methods, we propose a method for subspace clustering called Low-dimensional Localized Clustering (LDLC). In subspace clustering, data is partitioned into clusters, such that the points of each cluster lie close to a low-dimensional subspace. The unique property of LDLC is that it produces localized clusters on the underlying manifold of data. By conducting several experiments, we show this property is an asset in many machine learning tasks. This method can also be used for local dimensionality reduction. Moreover, LDLC is a suitable tool for forming the tesserae in TesseraMap, and also for creating the patches in IPA.

Halide perovskites have some unique advantages as optoelectronic materials. Metal halide perovskites have been attracting enormous attention for applications in optoelectronic devices such as photodetectors, light-emitting devices and field-effect transistors. The remarkable semiconducting properties have been intensively investigated in recent years. However, the performance of optoelectronics devices based on the conventional perovskite is limited by the ion migration, the mobility of the carriers and the light absorption in the near infrared region and so on. In a decade, numerous attempts are studied to further breakthrough the limitations using both physical and chemical methods. This dissertation is devoted to overcoming the drawbacks by integrating the state-of-art perovskite with other functional materials and to further deciphering the carrier transport mechanics behind the mixed dimensional heterostructures. Field-effect transistors are the workhorse of modern microelectronics. Proof-of-concept devices have been made, utilizing solution-processed perovskite as transistors. Beyond the Field-effect transistors, photodetectors can be construct with a transistor configuration. In this dissertation, we exploited Au dimers with structural darkness to enhance the light harvesting, and utilize sorted semiconducting single-walled carbon nanotubes to enhance the conductivity of thin-film. At last, we developed a hybrid memtransistor, modulable by multiple physical inputs using hybrid perovskite and conjugated polymer heterojunction channels to realize neuromorphic computing.

There are a variety of domains where it is desirable to learn a representation of an environment defined by a stream of sensori-motor experience. This dissertation introduces and formalizes subjective mapping, a novel approach to this problem. A learned representation is subjective if it is constructed almost entirely from the experience stream, minimizing the requirement of additional domain-specific information (which is often not readily obtainable). In many cases the observational data may be too plentiful to be feasibly stored. In these cases, a primary feature of a learned representation is that it be compact---summarizing information in a way that alleviates storage demands. Consequently, the first key insight of the subjective mapping approach is to phrase the problem as a variation of the well-studied problem of dimensionality reduction. The second insight is that knowing the effects of actions is critical to the usefulness of a representation. Therefore enforcing that actions have a consistent and succinct form in the learned representation is also a key requirement. This dissertation presents a new framework, action respecting embedding (ARE), which builds on a recent effective dimensionality reduction algorithm called maximum variance unfolding, in order to solve the newly introduced subjective mapping problem. The resulting learned representations are shown to be useful for reasoning, planning and localization tasks. At the heart of the new algorithm lies a semidefinite program leading to questions about ARE's ability to handle sufficiently large input sizes. The final contribution of this dissertation is to provide a divide-and-conquer algorithm as a first step to addressing this issue.

There are a variety of domains where it is desirable to learn a representation of an environment defined by a stream of sensori-motor experience. This dissertation introduces and formalizes subjective mapping, a novel approach to this problem. A learned representation is subjective if it is constructed almost entirely from the experience stream, minimizing the requirement of additional domain-specific information (which is often not readily obtainable). In many cases the observational data may be too plentiful to be feasibly stored. In these cases, a primary feature of a learned representation is that it be compact---summarizing information in a way that alleviates storage demands. Consequently, the first key insight of the subjective mapping approach is to phrase the problem as a variation of the well-studied problem of dimensionality reduction. The second insight is that knowing the effects of actions is critical to the usefulness of a representation. Therefore enforcing that actions have a consistent and succinct form in the learned representation is also a key requirement. This dissertation presents a new framework, action respecting embedding (ARE), which builds on a recent effective dimensionality reduction algorithm called maximum variance unfolding, in order to solve the newly introduced subjective mapping problem. The resulting learned representations are shown to be useful for reasoning, planning and localization tasks. At the heart of the new algorithm lies a semidefinite program leading to questions about ARE's ability to handle sufficiently large input sizes. The final contribution of this dissertation is to provide a divide-and-conquer algorithm as a first step to addressing this issue.

This thesis focuses on materials whose properties are directly linked with the dimensionality of their structures. We utilise diffraction techniques to characterise three systems. The first of these is an organic crystal, 2,3,6,7,10,11-hexahydroxytriphenylene tetrahydrate that contains one-dimensional water channels. These channels are polar and net polarity is achieved at low temperatures. The structure undergoes a phase transition to a non-polar state at higher temperatures. Mapping this finding onto an Ising model is undertaken in this thesis. We also investigate the possibility of transforming the same crystals to an organic metal with oxidation. Furthermore, this work characterises two-dimensional layered structures silver(I) tricyanomethanide and nickel(II) cyanide. We show that the former possesses the unprecedented property of negative area compressibility and the latter shows ordinary compressibility behaviour. Both these structures exhibit area negative thermal expansion. Finally, we investigate copper(II) and cobalt(II) doping into the structure of nickel(II) cyanide and demonstrate that the latter leads to a 10-fold increase in area negative thermal expansion.

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The need to renew and expand civil infrastructure, combined with an increased acknowledgement of a changing climate, has highlighted the need to incorporate the influence of climatic factors into the design of infrastructure. In geotechnical engineering, this includes understanding how climate influences the performance of slopes associated with engineered cuttings in pre- existing natural landforms. This understanding extends to both hydrological and hydrogeological conditions, both of which are often analyzed using numerical modeling of surface water and groundwater. Climate change predictions for Northern Ireland indicate that the amount and intensity of rainfall and extreme weather events will increase. This has raised concerns regarding the stability of existing engineered cut-slopes and the design of future highway and railway infrastructure. Recent studies have indicated that there is a link between pore pressure cycles and softening of slope structures, especially in clay rich materials typical of glacial till drumlins in Northern Ireland. These pore pressure fluctuations are caused by seasonal changes in the rate of recharge which then propagate through the deeper hydrogeologic system. As a consequence, the design of these cuttings requires that the hydrogeological response of these landforms to seasonal climate variations be incorporated into geotechnical designs. Two dimensional hydrogeological simulations are typically used in engineering practice. The main objective of this study was to evaluate the sensitivity of these simulations to dimensionality (two- and three-dimensions). The primary focus was on steady state groundwater flow within two drumlins with large slope cuts. Two- and three-dimensional groundwater models were developed using available information for a highway and a railway study site. The performance of each of these models was then compared to field monitoring from each site. A series of sensitivity studies were undertaken to evaluate the influence of key material properties and boundary conditions. Estimated recharge rates were found to range from 21 to 31 mm year-1 for both the railway (Craigmore) and highway (Loughbrickland) study sites. The hydraulic head distribution at the Craigmore site was similar for both dimensional simulations with a “best-fit” recharge rate of 50 to 60 mm year-1. At the Loughbrickland site, similar hydraulic head distributions with the “best-fit” recharge rate of 80 mm year-1 were reached in both dimensions. Overall, the research completed here emphasized the importance of gathering appropriate data prior to conducting development of hydrogeological models. As more data is made available, the overall complexity of the system can be better understood. As the complexity of the problem increases, the requirements for understanding the hydrogeological system in all three-dimensions becomes more important.

Data representations in low dimensions such as results from unsupervised dimensionality reduction methods are often visually interpreted to find clusters of observations. To identify clusters the result must be appreciably clustered. This property of a result may be called "clusteredness". When judged visually, the appreciation of clusteredness is highly subjective. In this paper we suggest an objective way to assess clusteredness in data representations. We provide a definition of clusteredness that captures important aspects of a clustered appearance. We characterize these aspects and define the extremes rigorously. For this characterization of clusteredness we suggest an index to assess the degree of clusteredness, coined the OPTICS Cordillera. It makes only weak assumptions and is a property of the result, invariant for different partitionings or cluster assignments. We provide bounds and a normalization for the index, and prove that it represents the aspects of clusteredness. Our index is parsimonious with respect to mandatory parameters but also exible by allowing optional parameters to be tuned. The index can be used as a descriptive goodness-of-clusteredness statistic or to compare different results. For illustration we use a data set of handwritten digits which are very differently represented in two dimensions by various popular dimensionality reduction results. Empirically, observers had a hard time to visually judge the clusteredness in these representations but our index provides a clear and easy characterisation of the clusteredness of each result. (authors' abstract) / Series: Discussion Paper Series / Center for Empirical Research Methods

This article provides a framework for assessing and quantifying "clusteredness" of a data representation. Clusteredness is a global univariate property defined as a layout diverging from equidistance of points to the closest neighboring point set. The OPTICS algorithm encodes the global clusteredness as a pair of clusteredness-representative distances and an algorithmic ordering. We use this to construct an index for quantification of clusteredness, coined the OPTICS Cordillera, as the norm of subsequent differences over the pair. We provide lower and upper bounds and a normalization for the index. We show the index captures important aspects of clusteredness such as cluster compactness, cluster separation, and number of clusters simultaneously. The index can be used as a goodness-of-clusteredness statistic, as a function over a grid or to compare different representations. For illustration, we apply our suggestion to dimensionality reduced 2D representations of Californian counties with respect to 48 climate change related variables. Online supplementary material is available (including an R package, the data and additional mathematical details).