An Explanation for the Logarithmic Connection between Linear and Morphological System Theory Abstract. Dorst/van den Boomgaard and Maragos introduced the slope transform as the morphological equivalent of the Fourier transform. Generalising the conjugacy operation from convex analysis it formed the basis of a morphological system theory that bears an almost logarithmic relation to linear system theory; a connection that has not been fully understood so far. Our article provides an explanation by disclosing that morphology in essence is linear system theory in specific algebras. While linear system theory uses the standard plus-prod algebra, morphological system theory is based on the max-plus algebra and the min-plus algebra. We identify the nonlinear operations of erosion and dilation as linear convolutions in the latter algebras. The logarithmic Laplace transform makes a natural appearance as it corresponds to the conjugacy operation in the max-plus algebra. Its conjugate is given by the socalled Cramer transform. Originating from stochastics, the Cramer transform maps Gaussians to quadratic functions and relates standard convolution to erosion. This fundamental transform relies on the logarithm and constitutes the direct link between linear and morphological system theory. Many numerical examples are presented that illustrate the convexifying and smoothing properties of the Cramer transform. Linear system theory is a successful and well established field in signal and image processing (Castleman, 1996; Gonzalez and Woods, 2002; Hamming, 1998; Oppenheim, 1999) . In the n-dimensional case, shift invariant linear filters can be described as convolutions of some signal f : IR n → IR with a kernel function b : IR n → IR: play an important role as convolution kernels: They are the only separable and rotationally invariant func- tions that preserve their shape under the Fourier transform. Convolutions of a signal f with the family {K σ | σ > 0} of Gaussians create the Gaussian scalespace (Iijima, 1959; Weickert et al., 1999; Witkin, 1983) , a multiscale representation that is useful in pattern recognition, image processing and computer vision (Florack, 1997; Iijima, 1973; Lindeberg, 1994; Sporring et al., 1997) . Figure 1 (a) shows an example. Mathematical morphology is an interesting nonlinear alternative to linear systems theory (Heijmans, 1994; Maragos and Schafer, 1990; Serra, 1982; Serra, 1988; Soille, 1999) . It has been applied successfully to a large number of fields including cell biology, computeraided quality control, mineralogy, remote sensing and medical imaging. Morphology is based on two fundamental processes: dilation and erosion. In the case of nonflat morphology, the dilation resp. erosion of some function f : IR n → IR with a structuring function b : IR n → IR can be defined as follows (see e.g. (Maragos, 1994; van den Boomgaard, 1992) ): Dorst and van den Boomgaard (1994) and Maragos (1994) developed independently and simultaneously a morphological system theory that closely resembles linear system theory. Following (Dorst and van den Boomgaard, 1994) , one may generalise the dilation to the tangential dilation via. where stat y f (y) := { f (z) | ∇ f (z) = 0} denotes the stationary values of f . Then the morphological equivalent to the Fourier transform is given by the slope transform. a transformation that is closely related to the conjugacy operation in convex analysis (Heijmans and van den Boomgaard, 2001 ). Its backtransformation is given by. The slope transform allows to replace the tangential dilation by simple addition in the slope domain:. are those structuring functions in morphological system theory that play a role comparable to Gaussians in linear system theory (van den Boomgaard, 1992): They are the only rotationally invariant and separable structuring functions that maintain their shape under the slope transformation. The corresponding dilation . and erosion scale-spaces are depicted in Fig. 1(b) and (c). For a detailed analysis of their scale-space properties, we refer to Jackway and Deriche (1996) . Morphological scale-spaces with paraboloids as structuring functions are useful for computing Euclidean distance transformations (van den Boomgaard, 1992), for image enhancement (Schavemaker et al., 2000) and for multiscale segmentation (Jackway, 1996) .. The connections mentioned above are displayed in Table 1 . This table suggests that there is an almost logarithmic connection between linear and morphological system theory. The structural similarities between linear and morphological processes have triggered Florack (2001) , Florack et al. (1999) and Welk (2003) to construct parameterised processes that incorporate Gaussian scale-space and both types of morphological scale-spaces as limiting cases. Heijmans and van den Boomgaard (Heijmans, 2001; Heijmans and van den Boomgaard, 2001 ) have investigated unifying algebraic definitions of scale-space concepts that include a number of linear and morphological approaches (cf. also (Alvarez et al., 1993) ).. However, in spite of these very interesting contributions, the reason for the almost logarithmic connection between linear and morphological systems is still not sufficiently well-understood. To address this problem is the topic of the present paper.. We provide an algebraic explanation for the structural analogies between linear and morphological systems by revealing that morphology in essence is linear system theory in a specific algebra. While classical linear system theory uses the standard plus-prod algebra, the morphological system theory is based on the max-plus algebra and the min-plus algebra. This allows us to identify the nonlinear operations of erosion and dilation as linear convolutions * e and * d induced by these algebras. In this sense, morphology may be regarded as linear system theory in disguise.. These algebraic structures have already numerous interesting applications in other fields (Baccelli et al., 1992) : so-called discrete event dynamic systems (DEDS) can be modeled as linear systems with respect to these algebras. Discrete event dynamic systems in this algebraic formulation are used to find shortest paths in networks or to solve scheduling and communication problems in abstract project management, for instance. They are also employed to analyse queuing systems, traffic flow and the performance of special array processors. To the best of our knowledge, however, no attempt has been made so far to tackle problems from image analysis with this special algebraic approach.. Our paper is organised as follows. In Section 2 we introduce the max-plus and min-plus algebras that will play a fundamental role for the analysis of morphological systems. In Section 3 we show that dilation and erosion are convolutions in these algebras. Important concepts from convex analysis, in particular the conjugacy operation, are explained in Section 4. The re-interpretation of the conjugacy operation, viewed as a morphological operation, leads us to its counterpart in linear systems theory: the logarithmic Laplace transform which is introduced in Section 5. It provides an explanation for the logarithmic connection between linear and morphological systems. Another logarithmic connection is established in Section 6, where we investigate a continuous transition between convolution and morphological operations. Section 7 is devoted to the Cramer transform as the conjugate of the logarithmic Laplace transform. It constitutes a homeomorphism between linear and morphological systems. We discuss its basic properties and present a number of examples. Finally we conclude our paper with a summary in Section 8.. A preliminary version of the present paper has been presented in Burgeth and Weickert (2003) . In this paper we have given an explanation for the almost logarithmic connection between morphological systems and linear systems: The conjugacy operation considered in the max-plus algebra corresponds to the logarithmic Laplace transform in the standard algebra. In fact, morphological systems become linear when viewed in appropriate algebras. We were able to identify corresponding notions: convolution ↔ erosion, logarithmic Laplace transform ↔ conjugacy operation, and Gaussians ↔ quadratic functions. The composition of the logarithmic Laplace transform and the conjugacy operation results in the Cramer transform. This fundamental transform gives the direct link between morphological and linear system theory.. The present article can be regarded as a step towards the unification of linear and morphological scale-space theory on the basis of a general linear system theory in an appropriate algebra. Taking full advantage of this connection may allow to translate results directly from one area to the other. This may trigger a more fruitful interaction of both paradigms that have evolved independently to powerful image processing tools. Finally, a unification within a more general algebraic framework may also help to identify novel image processing approaches that are based on other algebras. These points will be addressed in our future publications.

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ALGORITHMS FOR COMPUTING WITH NILPOTENT MATRIX GROUPS OVER INFINITE DOMAINS We develop methods for computing with matrix groups defined over a range of infinite domains, and apply those methods to the design of algorithms for nilpotent groups. In particular, we provide a practical nilpotency testing algorithm for matrix groups over an infinite field. We also provide algorithms to answer a number of structural questions for a nilpotent matrix group.The main algorithms have been implemented in GAP, for groups over the rational number field. In this paper we develop a technique for computing with matrix groups defined over infinite domains, based on changing the ground domain via congruence homomorphism. This technique has proved to be an efficient tool in linear group theory (see e.g. Dixon (1971, Chapter 3) ). It is particularly useful for handling finitely generated linear groups (see Wehrfritz (1973, Chapter 4) ), and affords a general approach to many computational problems for infinite matrix groups. We apply the technique here to algorithms for nilpotent matrix groups.. Let F be a field and let G ≤ GL(n, F) be given by a finite generating set. We obtain algorithms for carrying out the following tasks: and, if G is nilpotent, (ii) constructing a polycyclic presentation of G; (iii) testing whether G is completely reducible, and finding a completely reducible series of G-modules; (iv) deciding finiteness of G, and calculating |G| if G is finite;. (v) finding the p-primary decomposition of G, and finding all Sylow p-subgroups if G is finite.. These algorithms address standard problems in computational group theory (i, ii, iv) and computing with matrix groups (iii), and facilitate structural investigation of nilpotent linear groups (v).. Our guiding objective has been to design algorithms that cover the broadest possible range of infinite domains. However, for convenience or reasons of practicality we sometimes place restrictions on F. For example, a preliminary reduction in nilpotency testing assumes that F is perfect; and constructing polycyclic presentations requires F to be finite or an algebraic number field. Two important types of field that we treat throughout the paper are algebraic number fields and certain functional fields. Implementation and practicality of our algorithms depend on the machinery (such as polynomial factorization) that is available for computing with the various fields.. A finitely generated subgroup of GL(n, F) is contained in GL(n, R) for some finitely generated integral domain R ⊆ F. In turn, each completely reducible solvable subgroup of GL(n, R) is finitely generated (by Dixon (1971, Theorem 6.4, p. 111) and Wehrfritz (1973, 4.10, p. 57) ). Our algorithms therefore return information not only about finitely generated nilpotent subgroups of GL(n, F), but also about nilpotent subgroups of GL(n, R) for any finitely generated integral domain R. Furthermore, these algorithms may be regarded as a platform for computing in abstract finitely generated nilpotent groups, because such a group is isomorphic to a subgroup of GL(n, Z). A method for constructing a representation of a finitely presented polycyclic group in GL(n, Z) can be found in Lo and Ostheimer (1999) .. Nilpotency is an important group-theoretic property, and testing nilpotency is consequently one of the basic functions of any computational group theory system. This paper provides the first uniform and effective solution to the problem of computing with infinite nilpotent matrix groups. Our algorithms for nilpotency testing (over finite fields, and Q) have been implemented as part of the GAP package 'Nilmat' (Detinko et al., 2007) (this is joint work with Bettina Eick). Standard algorithms for nilpotency testing in GAP and MAGMA sometimes fail to decide nilpotency even for small finite matrix groups, and fail for almost all infinite matrix groups. In the paper's final subsection we give some experimental results, and other details of the 'Nilmat' package.

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Describing differences between databases We study the novel problem of efficiently computing the update distance for a pair of relational databases. In analogy to the edit distance of strings, we define the update distance of two databases as the minimal number of set-oriented insert, delete and modification operations necessary to transform one database into the other. We show how this distance can be computed by traversing a search space of database instances connected by update operations. This insight leads to a family of algorithms that compute the update distance or approximations of it. In our experiments we observed that a simple heuristic performs surprisingly well in most considered cases.Our motivation for studying distance measures for databases stems from the field of scientific databases. There, replicas of a single database are often maintained at different sites, which typically leads to (accidental or planned) divergence of their content. To re-create a consistent view, these differences must be resolved. Such an effort requires an understanding of the process that produced them. We found that minimal update sequences of setoriented update operations are a proper and concise representation of systematic errors, thus giving valuable clues to domain experts responsible for conflict resolution. Today, many scientific databases overlap in their sets of represented objects due to redundant data generation or data replication. For instance, in life science research it is common practice to distribute the same set of samples, such as clones, proteins, or patient's blood, to different laboratories to enhance the reliability of analysis results. Whenever overlapping data is generated or administered at different sites, there is a high probability of differences in results. These differences do not need to be accidental, but could be the result of different data production and processing workflows. For example, the three protein structure databases OpenMMS [5] , MSD [6] , and Columba [19] are all copies of the Protein Data Bank PDB [4] . However, due to different cleansing strategies, these copies vary substantially. Thus, a biologist is often faced with conflicting copies of the same set of real world objects and with the problem of solving these conflicts to produce a consistent view of the data.. Many inconsistencies are highly systematically due to different controlled vocabularies, different measurement units, different abbreviations, or due to consistent bias during experimental analysis. Learning about the reasons that led to inconsistencies is valuable information when assessing the quality of contradicting values to resolve conflicts. Unfortunately, in most cases only the final databases are accessible without any additional knowledge about the data generation or manipulation processes. Assuming that conflicts do not occur randomly, but follow specific (but unknown) regularities, patterns of the form "IF condition THEN conflict" provide valuable knowledge to facilitate their understanding. A domain expert could evaluate these patterns to assess the correctness of conflicting values and therefore for conflict resolution. In [16] we propose an algorithm for finding such patterns using association rule mining.. In this paper, we develop a different approach for describing "regular differences" among contradicting databases: The detection of minimal update sequences that transform one database into the other. We consider SQL-like update operations because these are the fundamental operations for the manipulation of data stored in relational databases. Given a pair of contradicting databases, each operation may (i) directly represent an update operation that has been applied to one of the databases if they both evolved from a common ancestor (as in the PDB example), or (ii) describe systematic differences between the databases. Consider the example in Figure 1 showing the fictitious results of two groups investigating the same set of owls (identified by their IDs). For transforming DBOwl into OWLBase Latin species names must be replaced with common names. We also need to modify the color for all snowy owls, as DBOwl uses a finer grained distinction in colors. Therefore, regardless of how these databases were generated, there are update operations that allow a user to infer possible reasons for the existing conflicts. We call the number of operations within a minimal update sequence the update distance between the two databases. Each update sequence as long as the update distance is one of the simplest possible explanations for the observed differences. Following the "Occam's Razor" principle, we conclude that the simplest explanations are usually the most likely. Minimal update sequences give valuable clues on what has happened to make a database different from its original state. Therefore, the update distance is a semantic distance measure, as it is inherently process-oriented in contrast to purely syntactic measures such as counting different values.. The update distance is defined analogously to the edit distance for strings [14] , i.e., the minimal number of edit operations that transform one string into another. In contrast to editing strings, setoriented database updates strongly influence each other thus giving importance to the order of operations. To give an idea of the complexity of the problem, consider transforming databases r 1 into r 2 of Figure 2 . A first approach solves each conflict individually ( Figure 2a ). The five resulting operations could be executed in any order. A greedy approach, solving as many conflicts as possible with each operation, results in a total of four update operations ( Figure 2b ). However, the minimal update sequence for transforming r 1 into r 2 has three update operations (Figure 2c ). Intuitively, it is often necessary to use operations that at first introduce new conflicts, because these conflicts can be used as discriminating conditions in later update operations. The first operation in Figure 2c temporarily increases the total number of conflicts, but this is compensated in later operations. 1 1 1 1 1 1 2 1 1 2 1 1 3 1 1 3 1 1 4 2 1 4 2 0 5 3 1 5 3 0 6 4 1 6 4 0 7 5 1 7 5 0 8 6 1 8 6 0 9 1 0 9 1 0 10 1 0 10 In this paper, we present exact and approximate algorithms for computing the update distance and for finding minimal sequences of update operations for a pair of databases. Even though we consider only a restricted form of updates (namely those where attribute values are set to constants), our algorithms for computing the exact solution require exponential space and time. However, we also present greedy strategies that lead to good results in all examples we considered.. The rest of this paper is structured as follows: The next section discusses related work. Section 3 gives our definition of update distance and derives an upper and lower bound, which are important for optimization. In Section 4, algorithms for finding minimal sequences are presented. Section 5 discusses fast approximate algorithms. In Section 6 we present experimental results, and Section 7 concludes our paper. There is also a longer version of this paper available for further details [17] . To the best of our knowledge the problem of finding minimal sequences of set-oriented update operations for relational databases has not been considered before. The main areas of related work are: definitions of distance for databases, representing differences of databases, and finding patterns in conflicting data.. There are other definitions of database distance based on the modification of databases. One comes from the area of computing consistent query answers for inconsistent databases [1] . Here, a repair for a database r violating a set of integrity constraints IC is defined as a database r' that satisfies IC and has minimal distance to r. In this context, the distance is defined as the number of tuples that are contained in one database, but not in the other. Thus, the distance definition is very different from ours, as we need to find the actual update sequences for transforming one database into the other. Bohannon et al. presented an algorithm for finding a repair that is the cheapest under a cost model for modifications and updates, also leading to a quite different notion of distance [7] . While their approach needs to consider the order of their operations they do not consider set-oriented update operations.. The latter is also true for other approaches that represent differences between databases. Chawathe et al. and Labioet et al. use sequences of insert, delete, and update operations to represent differences between database snapshots [13] or hierarchically structured data [8] . Both approaches consider only operations that affect a single tuple. Since databases are manipulated with (setoriented) SQL commands, we consider our problem as more natural than a tuple-at-a-time approach. Several other applications use so called "update deltas" to represent differences between databases. In database versioning they are used as memory effective representation of different database versions [10] . However, versioning collects the actual operations during execution instead of having to rediscover them from two given versions.. Other methods for finding patterns in contradictory data to support conflict resolution are presented in [16] and [11] . In [11] , the authors distinguish between context dependent and context independent conflicts. Context dependent conflicts represent systematic disparities that are consequences of conflicting assumptions or interpretations. Context independent conflicts are idiosyncratic in nature and are consequences of random events, human errors, or imperfect instrumentation. Accordingly, we consider only con-text dependent conflicts. However, in contrast to [11] , we do not consider complex data conversion rules for conflict resolution. Instead we always use one of the conflicting values as solution.. We consider the discovery of conflict conversion rules as future work in Section 7. However, we do consider the conflict causing context to be identifiable as data patterns. In [16] , we use a frequent itemset mining algorithm to find conflicts frequently occurring together with certain patterns in the data. These patterns can only provide a static view on the differences and do not take into account possible dependencies between conflicts. In the approach presented here, such dependencies are implicitly captured by the order of update operations within a (minimal) update sequence.. A prerequisite of our approach is the ability to identify identical real-world objects in different databases. In general, this is a difficult problem with a long history of research [12, 15, 22] that we consider as orthogonal to our problem. Throughout this paper, we assume the existence of a source-spanning object identifier. These identifiers may have been assigned to tuples by a preceding duplicate detection step. This section discusses some of the results of our experiments applying the defined algorithms to different pairs of databases. We implemented all algorithms using Java™ J2SE 5.0. The code and experimental data is available from the authors.

O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


RF interconnects for communications on-chip In this paper, we propose a new way of implementing on-chip global interconnect that would meet stringent challenges of coreto-core communications in latency, data rate, and reconfigurability for future chip-microprocessors (CMP) with efficient area and energy overheads. We discuss the limitation of traditional RC-limited interconnects and possible benefits of multi-band RF-interconnect (RF-I) through on-chip differential transmission lines. The physical implementation of RF-I and its projected performance versus overhead as the function of CMOS technology scaling are discussed as well. The CMOS technology scaling trend has been driven by the desire for higher performance and lower cost integrated circuits for several decades. Scaling enabled the decrease of digital gate capacitive loads, thereby reducing gate latency and also reduced the silicon area of digital gates, enabling the integration of more logic in the same chip area. Both trends contributed to microprocessor performance in terms of instructions performed per second, since lower latency enabled higher clock frequency, and more logic enabled more parallelism.. Two physical facts present significant challenges to the continued performance improvement via simple device scaling. One is dissipated power. As clock frequency increases, more power is consumed, up to a point where it causes serious thermal and reliability issues. Increased leakage, as a result of scaling, further cuts into the power budget, thus limiting the increase of frequency even more. The second physical fact is the scaling effect on the properties of metal wires that serve as device and unit interconnects. Previously regarded negligible, wire contribution to latency and power consumption became significant with scaling due to increased resistance and more significant capacitance compared with MOS devices. While local wires still scale nicely since their length is getting shorter, global wires used for communication across the chip are a potential inhibition to improving microprocessor performance.. One of the current trends to meeting the need for improved microprocessor performance with tolerable power consumption is chip multi processors (CMPs) [1] . Under this approach, scaling is used to reduce the area of processor cores while keeping the clock operation frequency relatively constant. Integrating more cores on the same chip area allows a performance benefit through parallelization. Communication between on-chip cores and shared on-chip caches can be done using the traditional bus. As the number of cores and caches banks further increases, the shared bus does not scale so easily, resulting in more latency for communication and consuming more power and area. An alternative architecture that seems to scale more easily is dubbed network-on-chip (NoC), in which routers are used in conjunction with every on-chip core and cache bank. In a simple mesh topology, each router can communicate with his four orthogonal neighbors and with his adjacent core or cache. Multi-bit repeated wire based buses are still assumed to connect the mesh-network routers. As a result, although close neighbors can easily communicate with high bandwidth and low latency in an NoC, communication between distant points across the chip incurs longer latencies. The bottleneck gets even more severe when overall chip communication is considered, when larger traffic increases router congestion and makes actual latencies even longer.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


On the Impact of Triangle Shapes for Boundary Layer Problems Using High-Order Finite Element Discretization The impact of triangle shapes, including angle sizes and aspect ratios, on accuracy and stiffness is investigated for simulations of highly anisotropic problems. The results indicate that for high-order discretizations, large angles do not have an adverse impact on solution accuracy. However, a correct aspect ratio is critical for accuracy for both linear and high-order discretizations. Large angles are also found to be not problematic for the conditioning of the linear systems arising from the discretizations. Further, when choosing preconditioning strategies, coupling strengths among elements rather than element angle sizes should be taken into account. With an appropriate preconditioner, solutions on meshes with and without large angles can be achieved within a comparable time. Many physical problems exhibit highly anisotropic behaviors, where the length scale in one direction can be several orders of magnitude smaller than in the other. To resolve these features, which often occur in thin boundary layers, wakes, or shocks, highly anisotropic elements need to be employed to improve computational efficiency.. Understanding the impact of triangle shapes can provide important guidelines for mesh generation and adaptation. While high aspect ratio meshes are important for efficiency to resolve anisotropic features, triangular elements with large angles are known to deteriorate solution quality. Babuška and Aziz demonstrated that the discretization error bound of a finite element approximation on triangular meshes grows without bound as the maximum angle of elements approaches 180. • asymptotically [3] . Although this is an asymptotic result and little can be said about the impact of large angles in a non-asymptotic range, avoiding large angles leads to the development of several mesh generation algorithms, including minimum-maximum triangulations [4, 15] , and hybrid methods with structured meshes in the regions where high anisotropy is needed [21, 22] . Both of these methods have different robustness and automation problems, especially for complex geometries, and avoiding large angles in mesh generation still remains a difficult problem.. Shewchuk conducted a comprehensive study on triangle shapes for linear finite elements [34] . He concluded that the triangle shape that leads to the best accuracy in terms of the H 1 -seminorm of interpolation error is one with an aspect ratio equal to the ratioκ of the maximum and minimum singular values of the solution Hessian but without large angles. However, he also demonstrated that this so-called "superaccuracy" is very fragile as the elements have to align precisely with the singular vectors of the Hessian, and so he suggested that generating triangles with an intermediate aspect ratio of √κ but possibly with large angles is a more realistic goal for anisotropic mesh generation. Also, Rippa showed in his work that this aspect ratio is optimal for the L p -norm of the interpolation error for 1 ≤ p < ∞ [29] . Although these results provide good insight into what an optimal triangle shape should be, their analysis is limited to linear elements. Recently, Cao developed an interpolation error bound involving anisotropic features of high-order derivatives, and proposed an optimal Riemannian metric for high-order interpolations [10, 11] . However, the impact of large angles is still unclear for high-order discretizations of highly anisotropic problems, and further, discretization errors can be much smaller than interpolation errors when large angles are present as argued for linear finite elements in [1] . If high-order discretizations are not sensitive to element angles, then large angles introduced by a Delaunay triangulation in a stretched space [8, 13] would not be problematic.. Another important aspect of numerical simulation is the solution efficiency of the linear system arising from the discretization applied. Many have investigated the relationship between mesh quality (e.g. element angle size) and the condition number of the stiffness matrix arising from a standard Galerkin approach for a selfadjoint elliptic problem. For example, Shewchuk derived a bound on the stiffness matrix condition number for standard linear finite element [34] , and recently in the work of Du et al. [14] , a refined bound was derived for a high-order finite element. However, little can be inferred from their work when convection terms are present, which make the stiffness matrix non-symmetric. Further, although stiffness matrix conditioning is important for the efficiency of iterative linear solvers, investigating preconditioning strategies for meshes with different geometric properties has a more practical value since for many applications, only the preconditioned systems are tractable.. In this paper, the impact of large angles and element aspect ratios will be studied for boundary layer problems using high-order discretizations. Both solution accuracy and conditioning of the discretized system will be investigated when large angles are present. Results show that using a high-order discretization, the same accuracy can be achieved at similar degrees of freedom for both large-angle and right-angle triangulations. However, a correct aspect ratio is critical for accuracy. It is also found that the two mentioned triangulations have similar linear conditioning, but solution schemes for these two should be different as they involve different couplings among elements. Incomplete LU factorization, being a popular preconditioning scheme, is investigated for effectiveness on both triangulations. This paper is organized as follows. Section 2 describes the discretization and the solution scheme applied. Sections 3 and 4 present the model problems and the meshes considered, including large-angle elements and right-angle elements. Section 5 studies the discretization errors on these meshes, and Section 6 studies conditioning of the linear systems arising from these meshes. In this work, the impact of triangle shapes, including large angles and aspect ratios, on accuracy and stiffness for simulations of highly anisotropic problems has been studied. With a high-order DG discretization, it is seen that the discretization error is not sensitive to the maximum angles involved, but using a correct aspect ratio is critical. In addition, the linear system induced from obtuse meshes has similar conditioning with that from acute meshes. However, the solution schemes for these two meshes differ as they involve different coupling strengths among elements. For this reason, ILU(0) has been shown less efficient on obtuse meshes for an isotropic diffusion problem. With an ILUT preconditioner, the solution on obtuse meshes can be achieved within a comparable time as on acute meshes. Therefore, for anisotropic meshing with highorder discretizations, while aspect ratio is critical, large angles can be overcome through small increases in preconditioning costs. For the block row corresponding to κ in the matrix K, the diagonal block K κκ involves the derivatives of all the three terms, while the off-diagonal block K κκ ′ has the derivatives of the terms II and III. Thus, with the blocks measured in a certain norm, the coupling weight of κ with κ ′ scales with . In other words, the coupling scales with O (l f /d κ ) 2 , where l f is the length of the face between κ and κ ′ , and d κ is a measure of the diameter of the element κ. For three-dimensional problems, the Tridiagonal Condition is still a sufficient condition for ILU(0) to be exact factorization, with an appropriate ordering regardless of the element angles involved. Further, consider an acute tetrahedron with a single short edge, and an obtuse tetrahedron in which all edges are of similar length but the height of the tetrahedron is much smaller. Then the acute tetrahedron has two faces much smaller than the other two, making the mesh very close to satisfying the Tridiagonal Condition. Hence, ILU(0) will work very efficiently on this mesh for an isotropic diffusion problem. On the other hand, the tetrahedra from the obtuse triangle will have all faces of similar areas, and so similar coupling strengths with neighbors. This will lead to a poor performance of ILU(0). In addition, ILUT might not be as efficient as in two-dimension because more off-diagonal blocks are present in each row of the Jacobian matrix. The conclusion about ILU(0) for the acute and obtuse elements is valid for a discretization with a nearestneighbor stencil or a discretization with the nearest-neighbor coupling much stronger than farther-neighbor coupling. Such discretization includes, for example, a DG discretization with BR2 for the diffusion term [5] , or a Compact DG method [26] . On the other hand, ILU(0) may be a poor preconditioner, even on the acute elements, for a discretization with more complicated coupling. For example, a standard Galerkin method has neighbor coupling defined among nodes, making ILU(0) less efficient than for a DG method.

O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Robust 1-bit Compressed Sensing and Sparse Logistic Regression: A Convex Programming Approach Abstract-This paper develops theoretical results regarding noisy 1-bit compressed sensing and sparse binomial regression. We demonstrate that a single convex program gives an accurate estimate of the signal, or coefficient vector, for both of these models. We show that an -sparse signal in can be accurately estimated from single-bit measurements using a simple convex program. This remains true even if each measurement bit is flipped with probability nearly 1/2. Worst-case (adversarial) noise can also be accounted for, and uniform results that hold for all sparse inputs are derived as well. In the terminology of sparse logistic regression, we show that Bernoulli trials are sufficient to estimate a coefficient vector in which is approximately -sparse. Moreover, the same convex program works for virtually all generalized linear models, in which the link function may be unknown. To our knowledge, these are the first results that tie together the theory of sparse logistic regression to 1-bit compressed sensing. Our results apply to general signal structures aside from sparsity; one only needs to know the size of the set where signals reside. The size is given by the mean width of , a computable quantity whose square serves as a robust extension of the dimension. Unlike traditional compressed sensing, which has already enjoyed an extraordinary wave of theoretical results, 1-bit compressed sensing is in its early stages. In this paper, we proposed a polynomial-time solver (given by a convex program) for noisy 1-bit compressed sensing, and we gave theoretical guarantees on its performance. The discontinuity inherent in 1-bit measurements led to some unique mathematical challenges. We also demonstrated the connection to sparse binomial regression and derived novel results for this problem as well.. The problem setup in 1-bit compressed sensing (as first defined in [5] ) is quite elegant, allowing for a theoretical approach. On the other hand, there are many compressed sensing results assuming substantially finer quantization. It would be of interest to build a bridge between the two regimes; for example, 2-bit compressed sensing would already open up new questions.

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Topic-bridged PLSA for cross-domain text classification In many Web applications, such as blog classification and newsgroup classification, labeled data are in short supply. It often happens that obtaining labeled data in a new domain is expensive and time consuming, while there may be plenty of labeled data in a related but different domain. Traditional text classification approaches are not able to cope well with learning across different domains. In this paper, we propose a novel cross-domain text classification algorithm which extends the traditional probabilistic latent semantic analysis (PLSA) algorithm to integrate labeled and unlabeled data, which come from different but related domains, into a unified probabilistic model. We call this new model Topic-bridged PLSA, or TPLSA. By exploiting the common topics between two domains, we transfer knowledge across different domains through a topic-bridge to help the text classification in the target domain. A unique advantage of our method is its ability to maximally mine knowledge that can be transferred between domains, resulting in superior performance when compared to other state-of-the-art text classification approaches. Experimental evaluation on different kinds of datasets shows that our proposed algorithm can improve the performance of cross-domain text classification significantly. In the traditional supervised learning framework, a classification task is to first train a classification model on a labeled training data. Then, the learned model is used to classify a test data set. Generally speaking, under such a framework, the learning algorithm relies on the availability of a large amount of labeled data. In practice, high-quality labeled data are often hard to come by, especially for learning tasks in a new domain. Labeling data in a new domain involves much human labor and is time-consuming. Fortunately, there are often plenty of labeled data from a related but different domain. This may be the case when the labeled data are out-of-date, but the new data are obtained from fast evolving information sources such as the Web blogs. The situation is more serious in current dynamically-changing Web environment. Unfortunately, traditional learning approaches cannot cope well with such a situation. For example, as described in [16] , a text classification model trained on a Yahoo! directory performs poorly on a Weblog classification problem, since the distribution of terms may differ significantly. Another example is newsgroup classification, where the news items for financial news may have a different distribution from the sporting events. It is often true that the performance of such classification decreases dramatically along the time dimension. This poses a new learning problem, because we often have to spend much effort in labeling the old documents, but it will cost us much to label the new ones. It would be a waste to throw away the old labeled documents entirely. In such a situation, how to accurately classify the new test data using as much of the old data as possible becomes a critical problem. In this paper, we focus on the problem of cross-domain text classification. We are given two data sets D L and D U , which are from two related but different domains. Here D L is a labeled data set from old domain, while D U is from a new domain and needs to be classified. We assume that, the class labels in D L and the labels to be predicted in D U are drawn from the same class-label set C. Under such circumstances, our objective is to accurately classify the documents in D U by fully utilizing the old domain data D L and their labels. We propose a novel approach called Topic-bridged PLSA (or TPLSA for short) for the cross-domain text classification problem. Our intuition is derived from the observation that the data in two domains may share some common topics, since the two domains are assumed to be relevant. Our key idea is to extend PLSA [8] to build a topic-bridge and then transfer the common topics between two domains. Using our TPLSA model, the common knowledge between two domains can be extracted as a prior knowledge in the model, and then can be transferred to the test domain through the bridge with respect to common latent topics. Overall, we combine this knowledge and the new knowledge learned from the unlabeled data for text classification in the test domain, even when the test domain has a different distribution from the training domain. In particular, we first extend the probabilistic latent semantic analysis (PLSA) [8] to incorporate both labeled and unlabeled data. Our extension allows us to use the hidden variables in PLSA as topics (or classes in classification setting) to bridge the documents in training and test domains, and learn under a joint probabilistic model. Such a new model is based on a simultaneous decomposition of the contingency tables associated with term occurrence knowledge in documents from both training and test domains, which identifies the principal topics of the training data as well as documents in the test data that support those topics. These topics are then taken as a bridge between the training and test domains. In addition, more prior knowledge from the training data is encoded in a set of constraints imposed between docu-ments, including the must-link constraints for documents to belong to the same cluster and cannot-link constraints for documents to belong to the different clusters. This prior knowledge is learned and applied for learning a classifier on the test data from different domains. According to the above algorithm, an object function is given involving both the likelihood on all the data and the pairwise constraints on the labeled training data. The EM algorithm is applied to iteratively maximizing such object function to acquire the final categories for the test data. A major advantage of our work is that by extending the PLSA model for data from both training and test domains, we are able to delineate nicely parts of the knowledge through TPLSA that is constant between different domains and parts that are specific to each data set. This allows the transferring of the learned knowledge to be naturally done even when the domains are different between training and test data. We conduct experiments to test the performance of TPLSA on 11 different datasets and compare our PLSA-based cross-domain text classification algorithm with other state-of-the-art algorithms. Experimental results prove that topic-bridged PLSA can achieve higher performance than other algorithms. The paper is organized as follows. In Section 2, we give a brief review of related work. In Section 3, topic-bridged PLSA is proposed for cross-domain text classification. The evaluation results are shown in Section 4. Section 6 concludes with a summary and suggestion for future work. In this paper, we proposed a novel algorithm called Topic-bridged PLSA to handle the cross-domain text classification problem by allowing knowledge learned from documents in one domain to be effectively transferred to another. The algorithm extends the traditional Probabilistic Latent Semantic Analysis (PLSA) to integrate the labeled training data and unlabeled test data under a joint probabilistic model with the common topic as bridge. We conducted experimental evaluation on 11 datasets and the results show that the proposed algorithm achieves better performance than other state-ofthe-art classification algorithms.. As a future work, we will consider other learning methods to acquire the parameters used in the TPLSA model, and consider other related classification tasks such as multi-class classification. Moreover, we plan to do further investigation to inspect the inherent relations among text datasets with different but related domains. The traditional classification learning assumes that the class labels are given for training data under the same distribution as the test data. Two schemes are generally considered, including supervised learning and semi-supervised learning. Supervised learning focuses on the case where the labeled data are sufficient. Naive Bayes classifiers [13] and support vector machines [11] are known as two of the most effective methods for supervised text classification. Semi-supervised learning [25] addresses the problem that the labeled data are too few to build a good classifier. It makes use of a large amount of unlabeled data, together with a small amount of the labeled data to enhance the classification. Examples for semi-supervised learning include EM-based methods [17] , transductive learning [12] etc. Both supervised and semi-supervised classification requires the labeled and unlabeled data should be under the same distribution. However, in our problem, the labeled and unlabeled data come from different domains, and their underlying distributions are often different from each other. This violates the basic assumption of traditional classification learning. Correcting sample selection bias [24] is another branch of work related to cross-domain classification. If we assume the domain discrepancy is only caused by sample selection bias, while other facts are ignored, we can simply apply the theory of correcting sample selection bias to solve the cross-domain classification problem. Sample selection bias [7] , the Nobel-prize winning work in Economics in 2000, assumes that the distribution difference is resulted from using non-randomly selected samples from the universe. Sample selection bias was firstly introduced in the econometrics [7] , and then came into the field of machine learning [24] . Zadrozny [24] proposed a two-step approach for correcting sample selection. The probability density is estimated to model the selection of training instances, and then the sample selection bias is corrected based on the estimated density. Several researches addressed the selection probability density estimation, e.g. kernel density estimation [20] and kernel mean matching [9] . Blitzer et al. [2] also analyzed the learning bounds for cross-domain learning based on instance weighting. However, those algorithms did not investigate the rich structure of the test data. Learning from auxiliary data [21] sources addresses classification using small amount of high quality base training data and large amount of low quality auxiliary training data. For cross-domain learning, we can consider the base and auxiliary training data come from different domains. In this area, Wu et al. [21] proposed an SVM-based algorithm using both base and auxiliary training data. They demonstrated some improvement by using the auxiliary data. After that, several other researches emerged to improve this learning task, e.g. active learning [14] , and boosting [5] . Different from these works, in our problem, there are not any base labeled data. The probabilistic latent semantic analysis [8] is a popular method for document clustering and related tasks. PLSA was derived from the well-known latent semantic analysis (LSA) for text analysis, and provides solid statistical foundation. In this model, each document is considered as the convex combination of several topics, where these topics or latent semantic variables are obtained using the maximum-likelihood principle. An expansion of PLSA is proposed in [4] , which integrates document content and hypertext connectivity for document clustering. Similar to other clustering methods, PLSA does not need labeled information, and thus does not consider the available prior knowledge of the domain. We adapt this model to include the labeled information on topics, thus arriving at a new model we call topic-bridged PLSA (TPLSA) that uses the training documents as source to extract the prior knowledge. By making use of this prior knowledge, our algorithm is able to find documents that relevant for each topic even when these documents are from different domains. In this paper, we will incorporate the must-link and cannot-link constraints into our TPLSA model, which borrowed the idea from semi-supervised clustering. Semi-supervised clustering [1] builds clusters under some additional constraints provided by a few labeled data, in the form of must-links (two examples must in the same cluster) and cannot-links (two examples cannot in the same cluster). It finds a balance between satisfying these constraints and optimizing the original clustering objective function. Several semi-supervised clustering algorithms have been proposed, including [1] [3] [10] . Our algorithm is essentially a classification algorithm in which the constraints given by the training data provide a class structure. It will be shown theoretically and empirically that our algorithm works well for cross-domain classification. In addition, we are interested in using the class-label knowledge gained from source domain training data to help classify documents in the target domain, which is not solvable by traditional semi-supervised clustering algorithms alone. In this section we empirically evaluate our algorithm for TPLSA for cross-domain text classification and compare it with other state-of-the-art algorithms.

B I O B B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I O O O O O O O


Mining Discriminative Patterns for Classifying Trajectories on Road Networks Abstract-Classification has been used for modeling many kinds of data sets, including sets of items, text documents, graphs, and networks. However, there is a lack of study on a new kind of data, trajectories on road networks. Modeling such data is useful with the emerging GPS and RFID technologies and is important for effective transportation and traffic planning. In this work, we study methods for classifying trajectories on road networks. By analyzing the behavior of trajectories on road networks, we observe that, in addition to the locations where vehicles have visited, the order of these visited locations is crucial for improving classification accuracy. Based on our analysis, we contend that (frequent) sequential patterns are good feature candidates since they preserve this order information. Furthermore, when mining sequential patterns, we propose to confine the length of sequential patterns to ensure high efficiency. Compared with closed sequential patterns, these partial (i.e., length-confined) sequential patterns allow us to significantly improve efficiency almost without losing accuracy. In this paper, we present a framework for frequent pattern-based classification for trajectories on road networks. Our comparative study over a broad range of classification approaches demonstrates that our method significantly improves accuracy over other methods in some synthetic and real trajectory data. W ITH recent improvements in GPS and RFID technologies, a tremendous amount of data about moving objects on road networks is being collected. The tracking of moving objects on road networks is becoming increasingly pervasive. GPS devices embedded in vehicles or other sensors on the streets can track a vehicle as it moves throughout the city traffic grid. A recent proposal from the Oregon Department of Motor Vehicles (DMV) suggests that a tracking system should be mandatory for all newly purchased and newly registered vehicles. Although privacy is a major concern of consumers, one study prepared for the US Department of Transportation says, "Less than 7 percent of the respondents expressed concerns about recording their vehicle's movements." One can expect that such trajectory data, without linking to private information, will become increasingly universal in the near future.. The importance of data analysis over trajectory data is being widely recognized [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] , [10] . This paper deals with the problem of classifying vehicles' trajectories on road networks. Trajectory classification is defined as the process of predicting a vehicle's class label based on its trajectory. Trajectory classification has many useful applications, such as city and transportation planning; road construction, design, and maintenance; traffic congestion recognition; law enforcement; and homeland security. In addition, it can be exploited for real-time monitoring [11] , e.g., for detecting suspicious movements of vehicles. Example 1. Intelligent transportation systems have become widespread in recent years. A popular objective of such systems is solving traffic congestion. Trajectory classification can facilitate traffic prediction [12] , which is one of the key technologies for solving traffic congestion. Suppose many trip paths (i.e., trajectories of vehicles) are stored in a database. Each of these trips is labeled with its destination, such as a district of a town or a city, and they are provided for training. The destination of a new trip is predicted by providing a partial trajectory to the trained classifier. Then, the amount of traffic at specific destinations can be predicted for the near future. If we know that an area will be congested in the near future, traffic can be rerouted.. Example 2. Activity recognition aims at recognizing the actions and goals of a person from a series of observations. The Opportunity Knocks prototype [13] is an example of activity recognition systems. A client transmits GPS signals to a server, and the server makes inferences about the behavior of the client (person) and gives suggestions on what to do next. The system is designed to assist cognitively impaired persons in finding their ways through city traffic. Trajectory classification can facilitate activity recognition since travel paths are closely related to activities. Here, the class label is an activity, e.g., commute, shopping, or leisure. Such classification is also useful for marketing since the server can send more appealing advertisements to users according to their current activities.. Feature-based classification has been widely used in the fields of data mining and machine learning [14] , [15] , [16] , [17] , [18] . In this paradigm, features are extracted from the data points, and each data point is transformed into a feature vector. Each feature vector represents the existence of features in its corresponding data point. These feature vectors are provided to a classifier such as the support vector machine (SVM) [19] for training and prediction.. One of the most important requirements for effective classification is discovering discriminative features. Thus, the following question naturally arises: "What are good feature candidates for trajectory classification?" Recently, the approach using frequent patterns for classification has been successfully adopted for relational data [14] , [15] , [20] , [21] , [22] , [23] , text documents [18] , protein sequences [17] , and graphs [16] . All of these studies demonstrate the usefulness of frequent patterns in classification. However, to the best of our knowledge, the question on trajectory data is yet to be answered.. To answer this question, we first analyze the behavior of trajectories on road networks. Intuitively, the locations where vehicles have visited are critical features for classification. However, we have found additionally that the order of these visited locations is crucial for improving classification accuracy. Based on our analysis, we assert that (frequent) sequential patterns [24] are good feature candidates since they preserve this order information. In experiments, we show that using sequential patterns improves classification accuracy significantly (by 10-15%) over a naive method that uses only individual road segments as features.. Furthermore, we address efficiency issues in trajectory classification because sequential pattern mining from trajectories tends to be time consuming. The length of sequential patterns is confined in the process of sequential pattern mining to enhance classification efficiency. Such length-confined sequential patterns are called partial sequential patterns. We show that, by using partial sequential patterns instead of closed sequential patterns [25] , feature generation time is improved significantly (by up to five times) with almost no accuracy loss.. In this paper, we present a framework of frequent patternbased classification for trajectories on road networks. Our framework consists of three steps: 1) feature generation, 2) feature selection, and 3) classification model construction. In the first step, partial sequential patterns are generated. In the second step, highly discriminative ones are selected for effective classification. In the third step, those selected patterns are fed into a classifier. The effectiveness of this framework is verified by comparing classification accuracy and efficiency with other alternatives.. In summary, the contributions of this paper are as follows:. . We analyze the behavior of trajectories on road networks and investigate good feature candidates for trajectory classification. . We apply frequent pattern-based classification [14] to trajectory classification. More specifically, we present a framework of using partial sequential patterns for trajectory classification. . We conduct a comparative study of a broad range of classification approaches.. . We demonstrate that our frequent pattern-based classification method significantly improves accuracy over other methods and that feature generation can be done efficiently owing to partial sequential patterns. The rest of the paper is organized as follows: Section 2 presents the problem statement. Section 3 discusses why partial sequential patterns are good feature candidates. Section 4 introduces our trajectory classification method based on partial sequential patterns. Section 5 presents the results of performance evaluation. Section 6 discusses related work. Finally, Section 7 concludes the study. In this paper, we have presented a framework of frequent pattern-based classification for trajectories on road networks. We have performed systematic analysis about the behavior of trajectories on road networks. Our analysis leads to the conclusion that partial sequential patterns are good feature candidates.. We have conducted extensive experiments by comparing classification accuracy and efficiency of five classification approaches. Experimental results show that Seq_DS improves classification accuracy by up to 15 percent over Single_All. At the same time, training time and prediction time are shown to increase only by less than 10 percent, and feature generation time is shown to decrease significantly with classification accuracy losing by as small as 1 percent. Also, in real data sets, Seq_DS achieves the highest classification accuracy among the five approaches. These results indeed demonstrate the effectiveness of trajectory classification using partial sequential patterns.. Overall, we believe that we have provided a practical framework for trajectory classification which can be used in a real environment. Also, we believe that the methodology developed here tells us that using sophisticated patterns can be quite useful in classification. Further study on efficient and effective methods for pattern-based classification with more sophisticated patterns is an interesting direction for future research. Our framework can be easily extended to use numerical attributes for classification. A set of attributes can be attached to each edge of a trajectory. Examples of such attributes include the average speed, top speed, elapsed time, day, timeof-day, etc. In this case, different single features are generated from one edge depending on the values of the attributes. 2. We expect that other feature selection methods will lead to similar results. In fact, SVM-Infoprop [30] using the weight vector of the SVM has shown to generate quite similar results.. 3. We show the definition for binary classification as in the original literature [29] . This definition can be easily extended to accommodate multiclass classification.. Suppose the time-of-day is an important factor for a specific classification application. Then, if a vehicle visits an edge e i at 9 a.m. and 6 p.m., these two visits should be represented by separate single features, e.g., ðe i ; morningÞ and ðe i ; eveningÞ.. An interesting issue is to decide a proper resolution for an attribute. In the above example, 9 a.m. can be represented by using various resolutions ranging from a high level to a low level, e.g., morning or 8-10 a.m. A proper resolution needs to be selected based on its discriminative power. Since there are more mature studies on this issue in feature generalization methods [8] , we can borrow their strategies and split the attribute domain using the selected resolution.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O B I O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I I O O B I O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Mechanizing common knowledge logic using COQ Abstract This paper presents a formalization in Coq of Common Knowledge Logic and checks its adequacy on case studies. Those studies allow exploring experimentally the proof-theoretic side of Common Knowledge Logic. This work is original in that nobody has considered Higher Order Common Knowledge Logic from the point of view of proofs performed on a proof assistant. As a matter of facts, it is experimental by nature as it tries to draw conclusions from experiments. Since one cannot be universal and know everything on everything, one must know something on everything. Indeed it is much more beautiful to know something on everything than to know everything on something; this universality is more beautiful. After the above statement, the proof is almost complete, but here I give complements for the interested reader that can be skipped in a first reading. Epistemic logic is usually mechanized by model checking [30, 31] . The work presented in this paper is to my knowledge the first exposition of the mechanization of the proof theory of Common Knowledge Logic based on rules. Concurrently Paulien de Wind has made her own mechanization using Coq based on an extension of natural deduction with several levels [43] using basically the above rule (Gen). It is close to reasoning in Kripke models and she has not investigated common knowledge. Notice that Common Knowledge Logic is more than modal logic. Not surprisingly there are many attempts to implement modal logic in logical frameworks. The LF group in Edinburgh has shown the difficulties of such an enterprise, with the clear conclusion that modalities are not easily coded in a logical framework [4, 5, 28] . Notice that I am not faced to that problem as I perform a deep embedding of a modal logic not just a coding of modalities in the logical framework. However the most noticeable papers exploring the connection between logical framework and modal logic are due to Basin, Matthews and Viganó [7, 8] . See there for a survey of the other approaches. Their implementation is not made in natural deduction, but in a modified natural deduction the so called labelled natural deduction for modal logic. Sequent calculus and natural deduction for Common Knowledge Logic has been studied by several authors [1, 22, 29, 38] , but none has studied Higher Order Common Knowledge Logic and none considers mechanization. For instance, Common Knowledge Logic in presence of induction or quantification has not been considered and even less quantification over propositions (see Section 4).

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On Searching a Table Consistent with Division Poset Suppose P n = {1, 2, . . . , n} is a partially ordered set with the partial order defined by divisibility, that is, for any two distinct elements i, j ∈ P n satisfying i divides j, i < Pn j. A table A n = {a i |i = 1, 2, . . . , n} of distinct real numbers is said to be consistent with P n , provided for any two distinct elements i, j ∈ {1, 2, . . . , n} satisfying i divides j, a i < a j . Given an real number x, we want to determine whether x ∈ A n , by comparing x with as few entries of A n as possible. In this paper we investigate the complexity τ (n), measured in the number of comparisons, of the above search problem. We present a Suppose P = {1, 2, . . . , n} is a partially ordered set (poset), we say a table A = {a i |i = 1, 2, . . . , n} of n distinct real numbers consistent with P , provided that for any i, j ∈ P satisfying i < P j, a i < a j . Given a table A of distinct real numbers which is consistent with a known poset P , and given a real number x, we want to determine whether x ∈ A, by making a series of comparisons between x and certain elements a i ∈ A. The problem is considered in a model using pairwise comparisons of the form x : a i (a i ∈ A) as basic operations. These comparisons have ternary outcomes x < a i , x = a i , or x > a i . Our aim is to make as few comparisons as possible. The complexity of the problem is defined to be the minimum, over all search algorithms for P , of the maximum number of comparisons required in the worst case.. In this article we consider the above search problem for the case where the partial order is defined by divisibility. Let P n = {1, 2, . . . , n} be a poset with the partial order such that for any two distinct elements i, j ∈ P n satisfying i divides j, i < Pn j. Thus we say a table A n = {a i |i = 1, 2, . . . , n} of distinct real numbers consistent with P n if for any two distinct elements i, j ∈ {1, 2, . . . , n} satisfying i divides j, a i < a j . Denote by τ (n) the complexity of the problem searching a given real number x in A n . We will investigate both upper bounds (i.e., search algorithms for A n ) and lower bounds on τ (n), our main result is the following

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Exploiting novelty, coverage and balance for topic-focused multi-document summarization Novelty, coverage and balance are important requirements in topic-focused summarization, which to a large extent determine the quality of a summary. In this paper, we propose a novel method that incorporates these requirements into a sentence ranking probability model. It differs from the existing methods in that the novelty, coverage and balance requirements are all modeled w.r.t. a given topic, so that summaries are highly relevant to the topic and at the same time comply with topic-aware novelty, coverage and balance. Experimental results on the DUC 2005, 2006 and 2007 benchmark data sets demonstrate the effectiveness of our method. Automatic document summarization has received great attention in recent years, due to the explosive growth of documents on the World Wide Web. One specific task of document summarization is topic-focused summarization, which is to generate summaries that are relevant to a given topic.. A good summary should satisfy the following three requirements: novelty ( [1] , also called diversity in [4] ), coverage [6, 4] and balance [4] . They require that sentences in a summary should describe different contents, cover every Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. CIKM '10 important aspect and pay more attention to more important aspects of the documents, respectively. When a topic is introduced for topic-focused summarization, we believe that topic-aware novelty, coverage and balance are preferred. These requirements are defined as follows. Topic-aware novelty requires that sentences in the summary should overlap less in topic-related contents. Topic-aware coverage requires that the summary should cover different aspects of topicrelated contents. Topic-aware balance requires that different aspects of topic-related contents should have the same relative importance in the summary as in the documents.. In this paper,we propose an unsupervised topic-focused multi-document summarization method. It differs from the existing methods in that the novelty, coverage and balance requirements are all modeled w.r.t. a given topic, so that summaries are obtained from sentences that are highly relevant to the topic and at the same time comply with topicaware novelty, coverage and balance. In the method, the relevance of a sentence to a topic is modeled as a topic relevance probability that takes into account the novelty requirement, while coverage and balance are modeled by a sentence selection preference function. Our method is quite effective. Extensive experiments on the DUC 1 2005 -2007 benchmark data sets demonstrate that our method outperforms representative existing approaches in all specified evaluation measures. We briefly mention some related summarization methods, especially the those that considers novelty, coverage and balance requirements.. Li et al. [4] treats summarization as a supervised sentence ranking problem, where structural SVM is applied. Novelty, coverage and balance requirements are incorporated into the sentence ranking process. However, [4] focuses on generic summarization, with no topic-related information considered. Unsupervised methods such as TextRank [5] , LexPagePank [3] and ManifoldRank [8] rank sentences based on graph ranking algorithms. [8] uses the maximum marginal relevance (MMR) [1] like method to penalize overlapping of sentences, where all words are treated equally whether or not they are related to the topic. The method proposed in [9] is closely related to our work in that it achieves topic-aware novelty. Similarity between sentences are biased toward the topic using the proposed query-sensitive similarity in the paper. Thus only overlapping of the topic-related contents is penalized if the MMR method is applied. Some summarization methods such as ClusterCMRW, ClusterHITS [7] and [6, 10] utilize clustering techniques. The rationale behind the clustering-based methods is that different clusters represent different aspects of the documents, which should all be covered if possible.

O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O