Improving phone duration modelling using support vector regression fusion In the present work, we propose a scheme for the fusion of different phone duration models, operating in parallel. Specifically, the predictions from a group of dissimilar and independent to each other duration models are fed to a machine learning algorithm which yields more precise phone duration predictions. We firstly study the performance of eight independent models. Here, among others, we investigate the effectiveness of Support Vector Regression (SVR), which has not yet been studied on the phone duration prediction task. At the fusion stage, we evaluate the performance of several regression algorithms. The evaluation was carried out on the American-English KED TIMIT and the Greek WCL-1 databases. On both databases, the SVR-based model demonstrated the lowest error rate.When compared to the second-best algorithm, we report a relative reduction of the mean absolute error (MAE) and the root mean square error (RMSE) by 5.5% and 3.7% on KED TIMIT, and 6.8% and 3.7% on WLC-1. The proposed fusion scheme, implemented with SVR fusion, improved the accuracy over the one of the best individual predictor, by 1.9% and 2.0% in terms of relative reduction of the MAE and RMSE on KED TIMIT, and by 2.6% and 1.8% on WCL-1 database. In this work we studied the accuracy of various machine learning algorithms on the task of phone duration prediction. The experimental results showed that on the task of phone duration prediction, Support Vector Machines (SVM), as a regression model, outperforms various other machine learning techniques. Specifically, in terms of relative decrease of the mean absolute error and root mean square error, the SMO regression model outperformed the second-best model by approximately 5.5% and 3.7% on KED TIMIT, and by approximately 6.8% and 3.7% on the WCL-1 database, respectively.. Furthermore, the proposed fusion scheme, which combines predictions from multiple independent phone duration models, operating on a common input, takes advantage of the observation that different prediction algorithms perform better in different conditions. The experimental validation demonstrated that the fusion scheme improves the accuracy of phone duration prediction. The SVM-based fusion algorithm was found to outperform all other fusion techniques. Specifically, the fusion scheme based on the SVM regression algorithm outperformed the best individual predictor (SVM regression) by approximately 1.9% and 2.0% in terms of relative reduction of the mean absolute error and root mean square error on the KED TIMIT database, and by 2.6% and 1.8% on the WCL-1 database, respectively.. 6. REFERENCES Fig. 1 . Block diagram of the proposed fusion scheme, which exploits multiple dissimilar phone duration predictors, operating on a common input. To investigate the practical usefulness of the proposed approach, we trained several individual phone duration models, and then employed them in the fusion scheme described in Section 2. The various individual phone duration models and the fusion algorithms involved in the phone duration prediction fusion scheme, as well as the speech databases used in the experiments and the experimental protocol that was followed, are described in the following subsections.

O O B I 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 B I I 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 B I I 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 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 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


Physical synthesis methodology for high performance microprocessors Integrated logic synthesis and physical design (physical synthesis) continues to play a very important role in high performance microprocessor design methodologies. In this paper, we present the integrated physical synthesis timing closure methodology used in the current generation microprocessors. Physical synthesis techniques were aggressively used as part of logic and placement optimizations for performance, power and area. The design turn around times were significantly reduced and timing convergence was consistently acheived. With technology scaling as well as the increasing complexity of modern microprocessors, interconnect effects play an increasingly important role in determining design methodologies. The design of the POWER4 [15] and of newer generations of other IBM high performance microprocessors [1] required considerable integration between synthesis and physical design to manage interconnect effects. The first generation of chips that used the physical synthesis methodology described in this paper, were fabricated in a 0.18-µm CMOS 8S3 SOI (silicon-on-insulator) technology with seven levels of copper wiring [11] . This version of the chip [15] had a clock frequency of greater than 1.3 GHz with a transistor count of 174 million. Physical synthesis continues to be used in other IBM microprocessors as well [1] . Acheiving timing closure on these chips continues to take significant innovations in physical synthesis techniques.. Microprocessor design entails very tight constraints on frequency, power, noise and packing density. Design closure and minimizing the effects of interconnect in the microprocessors while meeting tight performance constraints were accomplished by two main approaches: hierarchical design and physical synthesis with innovative techniques. This paper describes a physical synthesis methodology used in the design of microprocessors in a hierarchical design environment. Section 2 discusses a hierarchical approach for managing physical design, noise and timing issues while acheiving chip level design closure. The benefits of semicustom library design and how physical synthesis benefits from such libraries is described in Section 3. In Section 4, the physical synthesis methodology to accomplish macro level timing closure is described. Some post physical synthesis optimizations performed are described in Section 5. Section 6 presents some of the representative results that were obtained as part of the timing closure of the microprocessors. The paper presents a comprehensive physical synthesis methodology and techniques that have been used in the design of high speed microprocessors. The extensive use of physical synthesis has demonstrated the importance of integrating placement and synthesis tightly. Our experience has shown the need for further research in predicting and optimizing for interconnect and physical design effects much earlier in the design flow.

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


The complexity of Unique k-SAT: An Isolation Lemma for k-CNFs We provide some evidence that Unique k-SAT is as hard to solve as general k-SAT, where k-SAT denotes the satisfiability problem for k-CNFs with at most k literals in each clause and Unique k-SAT is the promise version where the given formula has 0 or 1 solutions. Namely, defining for each k 1, s k = inf{δ 0 | ∃ a O(2 δn )-time randomized algorithm for k-SAT} and, similarly, σ k = inf{δ 0 | ∃ a O(2 δn )-time randomized algorithm for Unique k-SAT}, we show that lim k→∞ s k = lim k→∞ σ k . As a corollary, we prove that, if Unique 3-SAT can be solved in time 2 n for every > 0, then so can k-SAT for all k 3.Our main technical result is an isolation lemma for k-CNFs, which shows that a given satisfiable k-CNF can be efficiently probabilistically reduced to a uniquely satisfiable k-CNF, with non-trivial, albeit exponentially small, success probability. While NP-complete search problems are presumably intractable to solve, this does not mean that all instances of these problems are difficult. Many heuristics for NP-complete problems, e.g., using back-tracking or local search techniques, take advantage of some structure in the instance. This leads to the general question: what makes an instance of a search problem easier than the worst-case? Conversely, we can ask, what types of structure cannot be exploited by heuristics? Here, we can consider both the structure of the instance (e.g. density of constraints to variables, or expansion of a formula when viewed as a hypergraph) and the structure of the solution space (e.g. how correlated are different solutions?).. One particularly natural measure of the solution space is the number of solutions. One can have different intuitions concerning how the number of solutions should affect the difficulty of the instance. On the one hand, one might suspect that the hardest instances would be those with few solutions, since they become "needles in a large haystack". On the other hand, algorithms that use local search or narrow in on the solution through deductions might do worse when there are many unrelated solutions. These algorithms might get caught between two equally attractive solutions, like Buridan's ass (who starves to death, unable to decide between two sources of food).. The best known algorithm for general CNF satisfiability (without any constraints on the clause size) is little better than exhaustive search, running in time O(2 n(1− 1 lg m ) ), where m is the number of clauses [Schuler03] . The situation is slightly better for k-SAT, where each clause of a given CNF contains at most k literals, for some constant k 3. This syntactic restriction can be exploited to obtain a O(2 k n )-time algorithm for k-SAT, for some constant 0 < k < 1 dependent on k. The first such algorithms were obtained in the 1980's [Dan81, MS85] . Recently, a number of faster algorithms have been proposed [PPZ99, PPSZ98, Sch02, HSSW02, DGH + ]. The best known randomized algorithm for 3-SAT is given in [HSSW02] . For k 4, the best known randomized algorithm is due to [PPSZ98] ; its time complexity on k-CNFs with n variables is O(2 (1−µ k /(k−1))n ), where µ k > 1 is an increasing function of k that approaches π 2 /6 ≈ 1.644.. The techniques of [PPSZ98] and its predecessor [PPZ99] crucially rely upon the notion of isolation of a satisfying assignment. The degree of isolation of a satisfying assignment x ∈ {0, 1} n is the number of coordinates 1 i n such that flipping the value of x i turns x into a falsifying assignment. Thus, a unique satisfying assignment has degree n of isolation, whereas any satisfying assignment of a tautology has degree 0 of isolation. The argument in [PPZ99] proceeds as follows. If a given k-CNF has "many" satisfying assignments, then we are likely to find one by random sampling. On the other hand, if there are "few" satisfying assignments, then there must be a satisfying assignment of "high" degree of isolation. However, sufficiently isolated satisfying assignments have "short" descriptions (via the Satisfiability Coding Lemma in [PPZ99] ), and so these assignments can be found by enumerating all possible "short" descriptions, and checking if any such description corresponds to a satisfying assignment. Combining the two cases yields a randomized algorithm of running time poly(n)2 (1−1/k)n . (Later, [PPSZ98] improved this algorithm.). A closer look at the analysis in [PPZ99] reveals that the more satisfying assignments a given k-CNF has, the faster we can find one of them. More precisely, as we prove later in the paper (see Section 4), if a k-CNF on n variables has s satisfying assignments, then one of them can be found with probability 1 2 in time poly(n)(2 n /s) 1−1/k . This result suggests that the hardest case of k-SAT may be Unique k-SAT, the promise problem whose positive instances are uniquely satisfiable k-CNFs, and whose negative instances are unsatisfiable k-CNFs.. In the rest of our paper, we provide further evidence towards the truth of this hypothesis.

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


A unified non-rectangular device and circuit simulation model for timing and power For 65nm and below devices, even after optical proximity correction (OPC), the gate may still be non-rectangular. There are several limited works on the device and circuit characterizations for the post-OPC non-ideal-shape wafer images, with significant impacts on timing and power. Most of them, however, are based on the equivalent gate length models, which are different for timing and leakage, and thus hard to use for coherent circuit simulations. In this paper, we propose a unified post-litho device characterization model and circuit simulation for timing and power. To our best knowledge, this is the most accurate methodology for post-litho analysis, including timing, leakage and transient simulation. Based on this method, the parameter extraction is also included in the model which was omitted by previous works. A post-litho model card is proposed for circuit simulation to combine these two techniques. Our experimental results validate the new model. As the IC technology evolves toward nanometer scale dimensions with higher densities, the printability of fine lithographic patterns with rectangular shapes is rapidly reduced due to the fundamental limit of both microlithography systems and process variations. As 193nm lithographic system is still used to print critical dimension (CD) of 65nm (and likely 45nm or even below feature size), various resolution enhancement techniques (RET), including optical proximity correction (OPC), phase shift mask (PSM), offaxis illumination (OAI) are used to push the lithographic systems to their limits. [1] Even with extensive RETs, the gate shapes may still be away from perfect rectangles which impact the timing of the whole chip [2] . Meanwhile, different process variations, such as dose, focus, etching variations, can push the manufactured silicon image (SI) of the gate even further away from the design-intended rectangle layout. In fact, as the channel region is determined by the interfaces between dopant profiles of channel and source/drain under the poly gate, the recent paper [3] has shown that even under rectangular gate, the channel region may be seriously still nonrectangular, which makes the rectangular gate effectively nonrectangular. As the continuous shrinkage of feature sizes, nonrectangular gates and channels are unavoidable due to the limitation of manufacturing process and have to be dealt with and simulated accurately.. The non-rectangular property of the gate SI has received a lot of research attention recently. It was first treated as random variations on the edges of the gate, i.e., line edge roughness (LER) [3] [4] [5] [6] [7] . The devices with LER can be simulated by 3D TCAD methods but they are too slow to be performed for circuit level simulation and optimization. A commonly used technique is gate slicing which applies 2D TCAD to study LER [5, 7] . As the non-rectangular of channel shape is most directly decided by the gate shape, the nonrectangular gate can be directly used to predict the channel shape. Recent works [8] [9] [10] use gate slicing and equivalent gate length (EGL) methods to simulate the impact of non-rectangular gate shapes in SPICE which is much faster than TCAD software. Two EGLs of each device are defined to replace the original uniform gate length set on layout: ON EGL for timing issues when the device is turned on and OFF EGL for leakage issues when the device is cutoff.. Although EGLs can model a non-rectangular device in either of its two specific working states (ON/OFF) well, they are hard to be used for coherent circuit simulations in practice, since it is often difficult to tell when and which devices are absolutely on or off for complicated cell schematics, thus it is hard to choose the right EGL for circuit simulation. Another limitation of these EGL works [8] [9] [10] is that they only studied the performance difference between non-rectangular and ideal rectangular gates. In fact, device models, such as BSIM3, BSIM4 [11] , and PTM [12] , and their parameters are extracted from the real devices with certain non-rectangular shapes. So the impact of gate shape with certain non-rectangles on device performance has been already included in the device models [10] . Performance simulations of gates between different nonrectangular shapes are necessary and crucial, or the impact of nonrectangular gate shape may be overestimated.. In this paper, we propose a novel unified non-rectangular device and circuit simulation model/methodology. Different from previous works of revalue specific parameter such as gate length, our work modifies the whole range of drain-source current model which is much more accurate than the two EGLs of ON and OFF. Meanwhile, the impact of nonrectangular gate shape during parameter extraction of device model (BSIM or PTM) can be fully considered. Moreover, as an additional device modeling card, it is based on I-V properties of the devices, and thus it can be integrated to any existing device models for post-lithography device/circuit simulation.. The rest of this paper is organized as follows: in section II, general model extraction flow including gate slicing is reviewed, with other preliminaries for our unified post-litho non-rectangular model. The details of post-litho device modeling card are discussed in Section III. Section IV shows how to integrate the modeling card into design flow and how to implement circuit simulation with the consideration of lithographic process variations. Simulation results are shown in Section V with the comparison between EGL and our unified model, as well as the impact of lithographic variations. An interesting observation is that in some cases, lithographic variations of defocus may be helpful in both timing and power. Conclusion is in the last section VI. In this paper, a new non-rectangular device characterization and circuit simulation methodology is proposed using the drain-source current modification. The impact of nonrectangular gate shape during parameter extraction of device model (BSIM or PTM) is considered in the first time. Our model is validated and compared to the existing equivalent gate length (EGL) methodology. Our simulation results show that EGL methods may lead to serious errors on both timing estimation (2%) and leakage/ power estimation (up to 1.5X). Our non-rectangular model provides a unified and accurate extraction, characterization, and simulation flow for both timing and power. Given that nanoscale devices are becoming more and more non-rectangular, we expect our unified model be very useful for accurate timing and power analysis in future nanometer designs. The simulation results are shown in this section. As the drain current of non-rectangular devices are various with different V gs and V ds , the uncertainty of EGL of different device working states are studied in part A. As mentioned in section II part D, the ON and OFF ELG are only accurate for some specific discrete working states, the post-litho device modeling card without considering SI EX is verified under those working states in part B.. After considering the non-rectangular gate shapes for device model parameter extraction, timing issues (delay and slew) and power issues (dynamic and static) of an inverter are discussed in part C, D and E. In part C, compared to our post-litho modeling card, about 2% underestimation of delay and slew of ON EGL method is observed. The precise simulations also show that lithographic process variations (especially defocus) can induce 15% difference on timing, and impact rising and falling delays differently.. In part D our test case shows that neglecting the impact of SI EX , the power estimation (dynamic and static) of EGL method can be seriously wrong (10% to 1.5X). And certain lithographic defocus can greatly reduce dynamic power consumption while slightly increase the static power. In part E, the power supplied directly from voltage source V dd is studied. Results show that EGL method can be seriously wrong (up to 30% for static power) and lithographic variations may greatly increase the power consuming/load supplied by V dd , which can induce more challenge on power distribution.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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


Simultaneous Truth and Performance Level Estimation Through Fusion of Probabilistic Segmentations Abstract-Recent research has demonstrated that improved image segmentation can be achieved by multiple template fusion utilizing both label and intensity information. However, intensity weighted fusion approaches use local intensity similarity as a surrogate measure of local template quality for predicting target segmentation and do not seek to characterize template performance. This limits both the usefulness and accuracy of these techniques. Our work here was motivated by the observation that the local intensity similarity is a poor surrogate measure for direct comparison of the template image with the true image target segmentation. Although the true image target segmentation is not available, a high quality estimate can be inferred, and this in turn allows a principled estimate to be made of the local quality of each template at contributing to the target segmentation. We developed a fusion algorithm that uses probabilistic segmentations of the target image to simultaneously infer a reference standard segmentation of the target image and the local quality of each probabilistic segmentation. The concept of comparing templates to a hidden reference standard segmentation enables accurate assessments of the contribution of each template to inferring the target image segmentation to be made, and in practice leads to excellent target image segmentation. We have used the new algorithm for the multiple-template-based segmentation and parcellation of magnetic resonance images of the brain. Intensity and label map images of each one of the aligned templates are used to train a local Gaussian mixture model based classifier. Then, each classifier is used to compute the probabilistic segmentations of the target image. Finally, the generated probabilistic segmentations are fused together using the new fusion algorithm to obtain the segmentation of the target image. We evaluated our method in comparison to other state-of-the-art segmentation methods. We demonstrated that our new fusion algorithm has higher segmentation performance than these methods. M ULTIPLE-template-based segmentation methods are widely used for the segmentation of medical images [1] - [7] . This type of algorithm has two distinct parts: nonrigid registration and fusion. Generally, in the first step, templates are registered to the target image and then the aligned templates and their corresponding label maps are used by a fusion algorithm to determine the segmentation of the target image.. The majority voting method is the simplest method for the fusion of the templates [8] . In this approach, at each voxel, the label with the largest number of votes is selected as the label of the target image. One of the drawbacks of this approach is that all of the votes are equally weighted. However, because of the limitations of the registration algorithms and also the potential uncapturable inter-individual anatomical differences between some of the templates and the target image, the templates may not perform equally well. Thus, to have a more accurate result, the performance of the templates should be considered in the fusion process. The challenge here is that the performance of the templates is not known a priori and needs to be estimated.. In the STAPLE algorithm [9] , an expectation-maximization (EM) algorithm is used to simultaneously estimate the hidden ground truth and to estimate the performance of the aligned templates. At each iteration of the EM, performances of the templates are estimated based on comparison to an evolving estimate of the reference standard segmentation and the new performance parameters are used to update the reference standard segmentation. There are a number of extensions of the STAPLE algorithm which use a similar methodology for the fusion [10] - [14] , [7] , [15] .. The local intensity similarity of the target image and the templates are also employed to estimate the local weight of the templates in the locally weighted fusion methods [1] , [2] , [16] , [3] , [4] , [6] , [17] . In this approach, the aligned label maps corresponding to the templates with higher intensity similarity to the target image have more weights in the fusion process. While in STAPLE-based methods, performance of the templates are directly estimated by estimating the ground truth, there has been no direct association between the intensity similarity of the template and the target image, and the performance of the templates in the locally weighted fusion algorithms.. Recently, using intensity images of the templates in the fusion process have been considered in two STAPLE-based algorithms [18] , [19] . The multi-STEPS algorithm uses local intensity similarity of the templates and the target image to locally select the templates in the STAPLE algorithm [18] . Nonlocal-STAPLE algorithm uses local similarity of the templates and the target 0278-0062 © 2013 IEEE image in the STAPLE algorithm based on a nonlocal means approach to improve the segmentation accuracy [19] .. To further improve the segmentation accuracy, two independent approaches can be employed. Improvement of the fusion strategy or improvement of the input segmentations fed to the fusion algorithm. To improve the fusion strategy, we need to estimate the template performance more accurately. This is the main feature of any fusion algorithm. It is obvious that even very powerful fusion strategies cannot accurately estimate the segmentation of the target image, if the information provided by the input segmentations is inadequate. Thus, improving the accuracy of the input segmentations generated from the templates will increase the accuracy of the fusion algorithm.. In many fusion algorithms, the aligned template segmentations have been used directly to infer the segmentation of the target image. Thus, one way to improve the accuracy of input segmentations is to use a more accurate registration algorithm in the alignment phase. However, to have a smooth and invertible transformation, a diffeomorphic image registration algorithm should be used which means the registration algorithm cannot in principle capture nondiffeomorphic inter-individual differences between the templates and the target image. Moreover, global optimization of the registration parameters is not feasible in a realistic registration problem and only a local optimum can be achieved. Furthermore, with a finite number of templates, it is not feasible to sample all anatomical variabilities and to feed the fusion algorithms with the information required to get the perfect segmentation of the target image. We propose instead to improve the target segmentation by using intensity and label map information. To this end, intensity and label map images of each one of the aligned templates are used to train a classifier. We use a local Gaussian mixture model (GMM) as the classifier. Each one of the trained classifiers is used to segment the target image. The output of each one of the classifiers is a probabilistic segmentation of the target image.. The generated probabilistic segmentations could be the input segmentations of STAPLE as the desired fusion algorithm. However, the STAPLE algorithm cannot be used for the fusion of the probabilistic segmentations as it requires a segmentation label at each voxel. One way to solve this problem is to use the label with the highest probability at each voxel. However, we may lose some accuracy in this process. The challenging question is how these probabilistic segmentations can be directly fed into a fusion framework to simultaneously estimate the hidden ground truth and the performance parameters of the segmentation generators. In [20] an algorithm was proposed and validated that achieved for the first time, effective performance weighted fusion of probabilistic labels. The introduced algorithm, called softSTAPLE, simultaneously estimates the performance of the segmentation generators and a reference standard segmentation from a collection of probabilistic segmentations of an image using an EM algorithm.. In this paper, we introduce a novel extension of the STAPLE algorithm, which uses an EM framework to simultaneously estimate the performance of the segmentations and the hidden ground truth from a collection of probabilistic segmentations which is computationally comparable with the original STAPLE algorithm. In general, the new fusion algorithm can be used for the fusion and evaluation of statistical classifiers, manual segmentations specified as confidence levels, or any set of probabilistic segmentations of a target image.. In this paper, we use our new fusion framework for the parcellation and segmentation of brain structures. We show that by using this approach, the performance of fusion is significantly improved compared to the classic STAPLE. In addition, we have compared our method to other state-of-the-art segmentation algorithms and have shown the superiority of our approach. The main contributions of our work are: development of a new fusion algorithm for the estimation of the reference standard segmentation and the performance parameters of segmentation generators from a set of probabilistic segmentations and estimation of probabilistic labeling using the intensity and label map images of the templates and the target image. The rest of the paper is organized as follows. Section II describes our novel fusion algorithm. Section II-F explains the proposed method for the calculation of probabilistic decisions based on the intensity and label information. Experimental data, procedures, and results are explained in Section III and conclusions are presented in Section IV. We have developed a new EM algorithm that simultaneously estimates the hidden ground truth and performance parameters from a set of probabilistic segmentations. The algorithm can be used for the fusion and evaluation of statistical classifiers, manual segmentations specified as confidence levels, or any set of probabilistic segmentations of a target image.. In this paper, we have used the algorithm for the parcellation and segmentation of brain structures. Intensity and label map images of each one of the aligned templates have been used to train a local GMM-based classifier. Each one of the trained classifiers has been used to segment the target image which generates a probabilistic segmentation of the target image. We then locally estimate template performance by comparison to a hidden reference standard. In this way, we have identified local regions where the template matches or does not match due to uncaptureable inter-individual anatomical dissimilarities between the target image and the template and also the errors due to intrinsic registration inaccuracies which are then fused.. As a result of this work, we have introduced two distinct fusion algorithms: PSTAPLE, which uses global performance parameters and is useful when the performance of the raters does not vary spatially; and local MAP PSTAPLE, which addresses the problem of spatial variability of the rater performance.. These new fusion algorithms were tested on two brain databases. These results, in turn, were compared to those of other state-of-the-art segmentation methods. We demonstrated that PSTAPLE and local MAP PSTAPLE have higher accuracy compared to other state-of-the-art methods described in the literature.. The algorithm we have described enables the fusion of probabilistic input segmentations. Many intensity and label fusion algorithms generate such probabilistic segmentations. Our fusion approach allows the characterization of the performance of such algorithms even in the absence of a reference standard segmentation.. Although we have used a local GMM-based classifier to generate the probabilistic segmentations, it may be possible to improve the accuracy of the segmentation by utilizing more sophisticated classifiers. This is one of the important aspects of the algorithm which needs further development. In the previous section, we described our framework for the fusion of probabilistic segmentations where we have used a local GMM for the calculation of the probabilistic decisions. The estimated probabilities are based on the intensity and label information of the templates and the intensity information of the target image. In this section, we test and evaluate our segmentation method using synthetic data and two brain databases: the databases of IBSR and NMM.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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


Weakly Supervised Learning of Interactions between Humans and Objects Abstract-We introduce a weakly supervised approach for learning human actions modeled as interactions between humans and objects. Our approach is human-centric: We first localize a human in the image and then determine the object relevant for the action and its spatial relation with the human. The model is learned automatically from a set of still images annotated only with the action label. Our approach relies on a human detector to initialize the model learning. For robustness to various degrees of visibility, we build a detector that learns to combine a set of existing part detectors. Starting from humans detected in a set of images depicting the action, our approach determines the action object and its spatial relation to the human. Its final output is a probabilistic model of the humanobject interaction, i.e., the spatial relation between the human and the object. We present an extensive experimental evaluation on the sports action data set from [1], the PASCAL Action 2010 data set [2] , and a new human-object interaction data set. H UMAN action recognition is one of the most challenging problems in computer vision. It is important for a wide range of applications, such as video indexing and surveillance, but also image search. It is a challenging task due to the variety of human appearances and poses. Most existing methods for action recognition either learn a spatiotemporal model of an action [3] , [4] , [5] or are based on human pose [6] , [7] . Spatio-temporal models measure the motion characteristics for a human action. They are, for example, based on bags of space-time interest points [3] , [8] , [9] or represent the human action as a distribution over motion features localized in space and time [5] , [4] , [10] . Pose-based models learn the characteristic human poses from still images. The pose can, for example, be represented by a histogram-of-gradient (HOG) [7] , [11] or based on shape correspondences [6] .. Our approach, in contrast, defines an action as the interaction between a human and an object. Interactions are often the main characteristic of an action (Figs. 9, 10, 11, and 14) . For example, the action "tennis serve" can be described as a human holding a tennis racket in a certain position. Characteristic features are the object racket and its spatial relation to the human. Similarly, the actions "riding bike" and "wearing a hat" are defined by an object and its relation to the human.. In this paper, we introduce a weakly supervised (WS) approach for learning interaction models between humans and objects from a set of images depicting an action. We automatically localize the relevant object as well as its spatial relation to the human (Figs. 9, 10, 11, and 14). Our approach is weakly supervised in that it can learn from images annotated only with the action label, without being given the location of humans nor objects.. Most related to our approach are the works of Yao and Fei-Fei [12] and Gupta et al. [1] , who also learn humanobject spatial interactions. However, these approaches operate in a fully supervised (FS) setting, requiring training images with annotated object locations as well as human silhouettes [1] or limb locations [12] . Another work by Yao and Fei-Fei [13] deals with a somewhat different formulation of the problem. Their goal is to discriminate subtle situations where a human is holding an object without using it versus a human performing a particular action with the object (e.g., "holding a violin" versus "playing a violin"). Note how this model requires manually localized humans both at training and testing time.. A recent work [14] models the contextual interaction between human pose and nearby objects, but requires manually annotated human and object locations at training time for learning the pose and object models. A previous work by the same authors [15] models spatial relations between object classes such as cars and motorbikes for object localization in a fully supervised setting.. Interactions are used to improve human pose estimation in [16] by inferring pose parameters (i.e., joint angles) from the properties of objects involved in a particular humanobject interaction.. Co-occurrence relations between humans and objects have been exploited for action recognition in videos by Ikizler-Cinbis and Sclaroff [17] . However, these relations are looser than what we propose as there is no spatial modeling of the interaction. This paper introduced a novel approach for learning human-object interactions automatically from weakly labeled images. Our approach automatically determines objects relevant for the action and their spatial relations to the human. The performance of our method is comparable to state-of-the-art fully supervised approaches [1] , [12] on the Sport data set of Gupta et al. [1] . Moreover, on the PASCAL Action Challenge 2010 [2] , it outperforms the best contestant (Everingham et al. [2] ) on classes involving humans and objects.. In future work, we plan to extend our approach to videos, where temporal information can improve the detection of humans and objects. Furthermore, temporal information can help to model variations within an action class and action sequences over time.

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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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


Coefficient Quantization in Banach Spaces Abstract Let (e i ) be a dictionary for a separable infinite-dimensional Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a 'finite alphabet'. We investigate several approximation properties of this type and connect them to the Banach space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c 0 . We also show that, for every ε > 0, the unit ball of every separable infinite-dimensional Banach space X contains a dictionary (x i ) such that the additive group generated by (x i ) is (3 + ε) −1 -separated and 1/3-dense in X. We begin with the problem which is discussed in this paper. Let (X, · ) be a separable infinite-dimensional Banach space and let (e i ) be a semi-normalized dictionary for X (i.e. (e i ) has dense linear span in X). For a given choice of N ∈ N, consider the problem of approximating an element x ∈ X by an element of the 'lattice'. In many situations (e.g. when (e i ) is a Schauder basis for X) each coefficient k i /2 N of an approximant from D N ((e i )) will be bounded by a constant that depends only on (e i ) and x . In this case the approximant will be chosen from a collection of vectors in D N ((e i )) whose coefficients are quantized by a 'finite alphabet'. We investigate two natural approximation properties. The first of these, which we call the Coefficient Quantization Property (CQP), is defined roughly as follows: For every prescribed tolerance there exists a quantization such that every vector x = i∈E a i e i in X, that can be expressed as a finite linear combination of dictionary elements, can be approximated by a quantized vector y = i∈E d i e i with the same (or possibly smaller) support E. Thus, for each ε > 0, there exists N such that for every finite E ⊂ N and for every x with support contained in E there exists y ∈ D N ((e i )) supported in E such that x − y ≤ ε. We say that (e i ) has the Net Quantization Property (NQP) if D N ((e i )) is an ε-net for X, so that the above holds without requiring the support of the approximant y to be contained in the support of x.. Precise definitions and some useful permanence properties are presented in Sect. 2. One of our main results (Theorem 2.4) is the perhaps surprising fact that quantization of the unit ball for some ε < 1 automatically implies quantization of the whole space. A consequence of this is that, given ε > 0, every separable infinite-dimensional Banach space X admits a dictionary (x i ) contained in its unit ball such that the additive group generated by (x i ) is (3 + ε) −1 -separated and 1/3-dense in X (Theorem 5.5), which is an infinite-dimensional version of a finite-dimensional result of Rogers [25] .. Several examples of bases with the CQP, including the Schauder system for C ([0, 1] ) and a class of bases for C(K), where K is a countable compact metric space, are discussed in Sect. 3. On the other hand, it is shown that the Haar basis for C(Δ), where Δ denotes the Cantor set, is not a CQP basis. It turns out that all of the natural examples satisfy a stronger form of the CQP which we call the Strong Coefficient Quantization Property (SCQP). Roughly, this means that the quantization of each coefficient can be an arbitrary δ-net, not necessarily a discrete subgroup of R.. This paper was motivated by results on approximating bandlimited functions by quantized coefficients [7] , and results on quantization of finite-dimensional frames (cf. [2, 3, 31] ), as well as a result obtained by Gowers [14] on the stabilization of Lipschitz functions on the sphere of c 0 .. W.T. Gowers [14] proved that every real-valued Lipschitz function f on the unit sphere of X stabilizes (i.e. all infinite-dimensional subspaces Y contain a further infinite-dimensional subspace Z such that f is essentially constant on the unit sphere of Z) if (and only if, by [22] ) X contains c 0 hereditarily. A key feature of his argument was the fact that the unit vector basis of c 0 has the CQP. The main results of this paper, as summarized in the Main Theorem (see below), yield an intimate connection between the CQP and containment of c 0 .. In [7] Daubechies and DeVore consider bandlimited functions, i.e. functions f ∈ L 2 (R) whose Fourier transformf has a fixed bounded support, say [−π, π] . Applying now the inverse transform one obtains, after a straightforward computation, the following expansion for f :. If f L ∞ ≤ 1 they showed that one can replace the coefficients (i.e. f (n/λ)) in the above expansion by quantized coefficients q n ∈ {−1, 1} so that. (1.1). The coefficients were obtained using a two-sided version of the Σ-Δ quantization algorithm (see Sect. 3.2) . In [2, 3, 31] the Σ -Δ quantization algorithm was applied to frames in a finitedimensional Hilbert space and the approximation estimates obtained there depend on the dimension of the space.. The result in [7] as well as the results in [2, 3, 31] raise the question whether or not it is possible to approximate given expansions with respect to a dictionary using quantized alphabets, without getting estimates which depend on the dimension of the space and without having to change the norm in such a way that the space under consideration contains c 0 isomorphically as in (1.1).. The main result in our paper excludes this possibility in the case of a fundamental and total normalized minimal system.

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


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database

technology.

Although

several

transformation

algorithms

exist,

they

are

incomplete

in

the

sense

that

they

focus

only

on

structural

aspects

and

ignore

semantic

aspects.

In

this

paper,

we

present

the

semantic

knowledge

that

needs

to

be

captured

during

transformation

to

ensure

a

correct

relational

schema.

Further,

we

show

an

algorithm

that

can

derive

such

semantic

knowledge

from

a

given

XML

Document

Type

De

nition

(DTD)

and

preserve

the

knowledge

by

representing

it

as

semantic

constraints

in

relational

database

terms.

By

combining

existing

transformation

algorithms

and

our

constraints-preserving

algorithm,

one

can

transform

XML

DTD

to

relational

schema

where

correct

semantics

and

behaviors

are

guaranteed

by

the

preserved

constraints.

Experimental

results

are

also

presented.

As

the

World-Wide

Web

becomes

a

major

means

of

disseminating

and

sharing

information,

Extensible

Markup

Language

(XML)

(Bray

et

al.,

is

emerging

as

a

possible

candidate

data

format

because

it

is

simpler

than

SGML,

and

more

powerful

than

HTML.

One

way

to

query

XML

data

is

to

reuse

the

established

relational

database

techniques

by

converting

and

storing

XML

data

in

relational

storage.

Since

the

hierarchical

XML

and

the

at

relational

data

models

are

not

fully

compatible,

the

transformation

is

not

a

straightforward

task..

To

this

end,

several

XML-to-relational

transformation

algorithms

have

been

proposed

(Deutsch

et

al.,

Florescu

and

Kossmann,

Shanmugasundaram

et

al.,

For

instance,

Shanmugasundaram

et

al.

presents

algorithms

that

focus

on

the

table

level

of

the

schema

while

Florescu

and

Kossmann

studies

di

erent

performance

issues

among

algorithms

that

focus

on

the

attribute

and

value

level

of

the

schema.

They

all

transform

the

given

XML

Document

Type

De

nition

(DTD)

to

relational

schema.

Similarly,

Deutsch

et

al.

presents

a

data

mining-based

algorithm

that

instead

uses

XML

documents

directly

without

a

DTD..

Although

they

work

well

for

the

given

applications,

they

miss

one

important

point.

That

is,

the

transformation

algorithms

only

capture

the

structure

of

a

DTD

and

ignore

the

hidden

semantic

constraints.

Consider

the

following

example..

Example

Consider

a

DTD

modeling

conference

publications:

<!ELEMENT

conf

(title,society,year,mon?,paper+)>

<!ELEMENT

paper

(pid,title,abstract?)>

Suppose

the

combination

of

title

and

year

uniquely

identi

es

the

conf.

Using

the

hybrid

inlining

algorithm

(explained

in

Section

the

DTD

would

be

transformed

to

the

following

relational

schema:

conf

(title,society,year,mon)

paper

(pid,title,conf_title,conf_year,abstract)

While

the

relational

schema

correctly

captures

the

structural

aspect

for

the

DTD,

it

does

not

force

correct

semantics.

For

instance,

it

cannot

prevent

a

tuple

t

from

being

inserted.

However,

tuple

t

is

inconsistent

with

semantics

of

the

given

DTD

since

the

DTD

implies

that

the

paper

cannot

exist

without

being

associated

with

a

conference

and

there

is

apparently

no

conference

yet.

In

database

terms,

this

kind

of

violation

can

be

easily

prevented

by

an

inclusion

dependency

saying

\paper

conf

title,conf

year]

conf

title,year]"..

The

reason

for

this

inconsistency

between

the

DTD

and

the

transformed

relational

schema

is

that

transformation

algorithms

only

capture

the

structure

of

the

DTD

and

ignore

the

hidden

semantic

constraints.

Via

our

constraintspreserving

inlining

(CPI)

algorithm,

we

show

the

kinds

of

semantic

constraints

that

can

be

derived

from

DTDs

during

transformation,

and

illustrate

how

to

preserve

them

by

rewriting

them

in

an

output

schema

notation.

Since

our

algorithm

to

capture

and

preserve

semantic

constraints

from

DTDs

is

independent

of

the

transformation

algorithms,

our

algorithm

can

be

applied

to

various

transformation

processes

such

as

Deutsch

et

al.

Florescu

and

Kossmann

Shanmugasundaram

et

al.

with

little

change.

Figure

presents

an

overview

of

our

approach.

First,

given

a

DTD,

we

transform

it

to

a

corresponding

relational

scheme

using

an

existing

algorithm.

Second,

during

the

transformation,

we

discover

various

semantic

constraints

in

XML

notation.

Third,

we

rewrite

the

discovered

constraints

to

conform

to

relational

notation..

This

paper

is

organized

as

follows.

Section

gives

background

information

and

related

work.

In

Section

the

transformation

algorithm

is

discussed

in

detail.

Section

presents

various

semantic

constraints

that

are

hidden

in

DTDs.

Section

proposes

our

algorithm

to

preserve

such

constraints

during

transformation.

Section

reports

some

experimental

results

and

Section

illustrates

two

example

applications

where

the

discovered

semantic

constraints

are

further

utilized.

Finally,

Sections

and

discuss

our

vision

on

future

work

and

concluding

remarks.

Due

to

many

bene

ts

from

using

relational

databases

as

storage

systems

for

XML

data,

the

need

for

e

cient

and

e

ective

conversion

between

relational

and

XML

models

will

signi

cantly

grow

in

a

foreseeable

future.

We

believe

the

following

directions

of

research

are

very

important..

First,

as

we

move

to

more

expressive

next

generation

XML

schema

languages

such

as

XML-Schema

(Fallside

(Eds),

or

RELAX

(Murata,

the

degree

of

complexities

captured

in

a

XML

schema

is

far

greater

than

that

in

a

DTD.

For

instance,

XML-Schema

supports

an

extensive

set

of

features

to

specify

structural

and

semantic

constraints.

However,

all

existing

XML

to

relational

conversion

algorithms

(discussed

in

Section

focus

only

on

the

DTD

case,

which

is

the

simpliest

and

least

expressive

XML

schema

language

according

to

Lee

and

Chu

Lee

et

al.

Therefore,

there

is

an

immediate

need

to

modify

and

extend

the

current

conversion

algorithms

to

support

more

complex

schema

languages..

Second,

with

XML

emerging

as

the

data

format

of

the

Internet

era,

there

is

a

substantial

increase

in

the

amount

of

data

encoded

in

XML.

However,

the

majority

of

everyday

data

is

still

stored

and

maintained

in

relational

databases.

Therefore,

we

expect

the

needs

to

convert

such

relational

data

into

XML

documents

to

grow

substantially

as

well.

Although

commercial

database

vendors

already

support

tools

that

generate

XML

documents

out

of

relational

data,

the

types

of

XML

documents

generated

are

very

simple

in

their

structure

and

consequently

cannot

capture

all

semantics

in

the

original

relational

schema.

For

instance,

a

majority

of

tools

can

only

convert

the

so-called

at

translation"

where

a

table

t

and

columns

c

i

of

relational

model

is

mapped

to

an

element

e

and

its

attributes

a

i

of

XML

model.

We

have

proposed

the

\nesting-based

translation"

to

capture

certain

semantics

in

the

original

relational

schema

however,

more

research

e

orts

in

that

direction

are

needed.

This

paper

presents

a

method

to

transform

XML

DTD

to

relational

schema

both

in

structural

and

semantic

aspects.

After

discussing

the

semantic

constraints

hidden

in

DTDs,

two

algorithms

are

presented

for:

discovering

the

semantic

constraints

using

the

hybrid

inlining

algorithm,

and

rewriting

the

semantic

constraints

in

relational

notation.

Our

experimental

results

reveal

that

constraints

can

be

systematically

preserved

during

the

conversion

from

XML

to

relational

schema.

Such

constraints

can

also

be

used

for

semantic

query

optimization

or

semantic

caching..

Despite

the

obstacles

in

converting

from

XML

to

relational

models

and

vice

versa,

there

are

several

practical

bene

ts:.

Considering

the

present

market

that

is

mostly

dominated

by

RDB

products,

it

is

not

easy

nor

practical

to

abandon

RDB

to

support

XML.

It

is

very

likely

that

industries

would

be

reluctant

to

adopt

the

new

technology

if

it

does

not

support

the

existing

RDB

techniques

as

they

were

reluctant

towards

object-oriented

database

in

the

past.

By

using

RDB

as

an

underlying

storage

system,

the

mature

RDB

techniques

can

be

leveraged.

That

is,

a

vast

number

of

sophisticated

techniques

(e.g.,

OLAP,

Data

Mining,

Data

Warehousing,

etc.)

developed

for

RDB

can

be

applied

to

XML

data

with

minimal

changes.

The

integration

of

a

large

amount

of

XML

data

on

the

Web

with

the

legacy

data

in

relational

format

is

possible.

Conversion

between

di

erent

models

has

been

extensively

investigated.

For

instance,

Christophides

et

al.

deals

with

transformation

problems

in

the

OODB

area;

since

OODB

is

a

richer

environment

than

RDB,

their

work

is

Table

Classi

cation

of

schema

conversion

between

XML

and

relational

models.

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


The Derivational Complexity Induced by the Dependency Pair Method We study the derivational complexity induced by the dependency pair method, enhanced with standard refinements. We obtain upper bounds on the derivational complexity induced by the dependency pair method in terms of the derivational complexity of the base techniques employed. In particular we show that the derivational complexity induced by the dependency pair method based on some direct technique, possibly refined by argument filtering, the usable rules criterion, or dependency graphs, is primitive recursive in the derivational complexity induced by the direct method. This implies that the derivational complexity induced by a standard application of the dependency pair method based on traditional termination orders like KBO, LPO, and MPO is exactly the same as if those orders were applied as the only termination technique.

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


Dynamically-optimized context in recommender systems Traditional approaches to recommender systems have not taken into account situational information when making recommendations, and this seriously limits the relevance of the results. This paper advocates context-awareness as a promising approach to enhance the performance of recommenders, and introduces a mechanism to realize this approach. We present a framework that separates the contextual concerns from the actual recommendation module, so that contexts can be readily shared across applications. More importantly, we devise a learning algorithm to dynamically identify the optimal set of contexts for a specific recommendation task and user. An extensive series of experiments has validated that our system is indeed able to learn both quickly and accurately. A recommender is a system capable of ranking a list of similar items with respect to one or more given criteria. There MDM 2005 05 Ayia Napa Cyprus (c) 2005 ACM 1-59593-041-8/05/05....$5.00 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. are many applications of recommender systems, notably in the information retrieval field, where criteria are submitted as queries and the most relevant documents are returned. With the proliferation of mobile E-services, recommenders in various forms have been the subject of many researches due to their obvious commercial values (e.g. [23] ). The two main recommendation techniques in use today are the content-based approach and the collaborative filtering approach [17] .. In content-based recommendation, the system suggests to users items that best fit a specified set of criteria. The system has to understand all the main features describing the items of interest in order to rank them in terms of their relevance to the criteria. Such systems typically maintain a notion of user profile that is highly specific to the problem domain e.g. profiles for research-paper recommenders typically reflect only the user's research interests, although the sharing of profiles among autonomous agents has been suggested [18] . On the other hand, collaborative filtering works by seeking the opinions of a community of users to assist individuals in that community to better identify contents of interest from a potentially overwhelming set of choices [19] . It relies on majority judgment within the community, and is largely based upon the notion of stereotypical behaviors and common interests. Hybrid systems exist today that use combinations of these two techniques to improve performance.. These traditional approaches suffer from a serious shortcoming: the set of features for consideration is static (fixed at design time) and entirely task-specific in nature. For instance, a user's query on a restaurant recommender could be "restaurants with vegetarian food ". A conventional recommender would simply rank all the known restaurants based on whether vegetarian food is available. Unknown to the application designer, and hence to the recommender, is that it is raining outside and the user would have preferred a nearer eating place. Clearly the system is not able to provide the best recommendations due to its detachment from the current situation; what is lacking in these traditional approaches is an awareness of the contexts. According to [7] ,. [Context is] any information that can be used to characterize the situation of an entity, where an entity can be a person, a place, or an object relevant to the interaction between the user and application, including the user and ap-plication themselves. A system is context-aware if it uses context to provide relevant information and/or services to the user, where relevancy depends on the user's task.. Over the years, many researchers have actively tried to realize the potentials of context-awareness e.g. in the more recent field of context-based information retrieval [15] . However, previous work on context-aware computing mostly support only static context, e.g. [1, 6, 10, 11, 12, 16, 22] . In all these applications, the composition of context is fixed at design time and no attempt is made to dynamically reduce or expand this set of contexts via learning through usage. In view of this, we propose a generic framework that enhances the existing recommender systems with a taskoriented, context-aware model. Our main contributions are described as follows.. • We advocate a formal distinction between context instantiation and recommendation. Separation of these concerns via an interactive, component-oriented design allows for a greater flexibility and clarity in roledivision. As such, our framework is generic and readily applicable to diverse problem domains. For example, an instantiation module can serve multiple different recommenders simply by plugging in appropriate context taxonomies. This modular design promotes efficiency and frees developers to focus on the task at hand.. • We propose the dynamic optimization of contexts for making recommendations to particular users. Systems would benefit from an intelligently-selected set of contexts in a myriad of ways. For example, the application designer may have specified contexts not regarded as relevant by a user. So our system would prune away the least relevant contexts while expanding it with more relevant ones overlooked by the designer. By instantiating only the most significant set of contexts, recommendation resources and time would be saved and the results would be more accurate.. The remainder of this paper is organized as follows: we present the details of our generic framework in Section 2, followed by the description of a sample implementation to realize this framework in Section 3. Section 4 presents our findings from experimental evaluations of this implemented system. In section 5, we compare our research to a number of related work. A discussion on some possible directions for future work in Section 6 concludes the paper. Figure 1 illustrates our proposed framework for domainrelevant context-aware recommendation systems. The modules within the framework are described below. Conventional recommender systems do not consider situational information and this seriously limits the relevance of their results. This paper advocates context-awareness as a promising approach to enhance recommenders' performance. We present a framework that separates contextual concerns from recommendation, so that contexts can be readily shared across applications. More importantly, we devise a learning algorithm that dynamically optimizes the context set for a specific recommendation task and user, and validate through extensive experiments that our system is capable of learning quickly and accurately.. We are extending this work in several directions. Firstly, we would like to investigate the possibilities of meaningfully applying a user's contextual preferences learned for a particular recommender to diverse problem domains in other systems. Along this line, we consider the extension of our approach to modelling groups of users instead of just individuals as an interesting direction.. Secondly, the issue of interdependencies among contexts should be resolved. The status of certain contexts may affect the importance of other contexts during decision-making. For example, the location of a restaurant should gain a greater importance when the weather is bad.. Thirdly, our restaurant recommender has been implemented as a desktop application. We would like to experiment with deployment on a mobile platform like the Smartphone and the PDA in order to study the user interaction issues.. Finally, we have investigated the dynamic reduction of the context set for recommendation considerations. It would be useful to be able to also dynamically expand the set of contexts for improving the recommendation performance. Dunlop et al described in [9] a palmtop application CityGuide for restaurant recommendation based on the match of restaurant types to a user's past preferences as well as ratings given by reviewers with similar preferences. They depended on the two user-specified filters of food type and price for profiling, and did not employ extensive learning to optimize the context set as what we have done in this work.. Tung et al demonstrated in [23] a prototype design of a software agent that was capable of recommending travel-related information based on the contexts of a user. Their recommendation procedure involved a dialogue between the user and the agent for modifying constraints given to the agent. The initial constraints were entered manually by the user as his or her preferences (e.g. budget, food-type, atmosphere, smoking or non-smoking). If a recommendation that fully satisfied all these constraints did not exist in the system's restaurant directory, the agent tried to relax the constraints one at a time and proposed these changes to the user, who had a choice of accepting any one of the proposed relaxations. The one restaurant in the directory that best fitted the relaxed set of constraints was then presented. Their system worked only with a statically defined set of contexts which could not be altered or optimized through usage.. A separate paper by Brunato et al proposed a middleware layer that collected a historical database of user position and URL usage information and then analyzed these to discover the links' spatial usage patterns [2] . A preference metric that reflected where and how often each link had been accessed by previous users was then computed. URLs were then ranked and recommended to the users based on their current location. Like us, they suggested the implicit gathering of usage feedback through user behaviors, e.g. whether any link was clicked, and which link was clicked. However, they considered only the current user location for recommendation, unlike our work which is much more comprehensive and can be readily applied to any contexts that are relevant to the recommendation task.. To the best of our knowledge, the only research that explicitly attempted to employ user modelling techniques within the domain of context-aware computing were the recent series of works by Byun and Cheverst [4, 5] . They argued that the user's preferences could be readily induced from the context history using user modelling and machine learning techniques, and that these modelled behaviors could be used together with the current contexts in supporting proactive environmental adaptations. In [5] , the authors described an experiment to demonstrate the learning of user's preferences from the context history for controlling an office environment. They defined a small static set of contexts, and collected the changes in these contexts as a time-stamped context history. From this history, they induced rules for representing the user's preferences regarding the status of window in various situations. They however did not make any attempt to dynamically optimize the user contexts as what we have proposed and extensively investigated.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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


Functional dependencies distorted by errors A relational database D is given with Ω as the set of attributes. We assume that the rows (tuples, data of one individual) are transmitted through a noisy channel (or, as many times in case of the datamining applications, the observed data is distorded from the real values in a manner which we cannot know). In case of low probability of the error it may be supposed that at most one data in a row is changed by the transmission or observation. We say that A → b (A ⊂ Ω, b ∈ Ω) is an error-correcting functional dependency if the data in A uniquely determine the data in b in spite of this error. We investigate the problem how much larger a minimal error-correcting functional dependency can be than the original one. We will give upper and lower bounds showing that the can be considerably larger than the original sizes, but the growth is only polynomial. Let us start with an example. Suppose that the pair of attributes (first name, last name) is a key in a database M (where the values in M are the real data). However, some of the data can be erroneous: the information is misunderstood in a phone conversation, the typist makes a mistake or the informant simply lies. We will denote by M * the database containing the available, observed, and so sometimes erroneous data. Note that in most of the cases below we will only assume the existence of the real, error-free database M , but will be able to use only the observed version, M * . Say, if we have both Maria Sklodowska and Mario Sklodowska in M and the first name "Mario" is replaced by "Maria" (M contains "Mario", M * contains "Maria"), we might have two individuals with names Maria Sklodowska in M * , hence the individual (row) cannot be determined from these two attributes. The question raised here is what other additional attributes we need to make us able to determine the real person (row).. A database can be considered as an m × n matrix M , where the rows are the data of one individual, the data of the same sort (attributes) are in the same column. Denote the set of attributes (equivalently, the set of columns of the matrix) by Ω, its size is |Ω| = n. It will be supposed that the data of two distinct individuals are different, that is, the rows of the matrix are different. Let A, B ⊂ Ω. We say that B functionally depends on A and write A → B if any two rows coinciding in the columns of A are also equal in the columns of B. Specially, if K → Ω then K is called a key. In other words, there are no two distinct rows of the matrix which are equal in K. A key is a minimal key if no proper subset of it is a key. Denote the family of all minimal keys by K.. Let M denote the matrix of the real data. These data are transmitted through a noisy channel. M * (m × n, again) denotes the matrix of the data obtained after the transmission. We will assume that the probability of errors is small and therefore that M and M * can differ in at most e entries in each row. Although it is also supposed here that the real data of two distinct individuals are different, that is the rows of M are different, the same cannot be stated about M * .. There might be several different reasons why we have to handle the erroneous database M * . If the sender (possessing and sending M ) and the receiver (receiving M * ) have both the possibility and the intention to cooperate, then the sender encodes the rows of M by an e-error-correcting code and sends the encoded row. The receiver decodes the received sequence and can recover the original row of M . That is, the receiver does not have, does not know any M * . This is not our case, we assume, that the sender or original owner of the database does not cooperate with the receiver/user; for example, in case of datamining applications, the observed data is often distorded from the real values, which we cannot know. Or in case the data is transmitted via a channel, an error can occur during transmission. There might also be intentional or unintentional mistakes during providing the data to the user(s). That is, the receiver cannot recover the original row. Our only possible goal is to find connections between the structure of M and the structure of M * , more precisely, between the system of functional dependencies of M and that of M * .. Two different models will be considered in the present paper. In Model 1 it is assumed that the system of functional dependencies of M is known by the receiver, we only want to find a connection between this and the system of functional dependencies defined by M * . On the other hand, in Model 2 nothing is known about the system of functional dependencies in M , we only know the received rows of M * and our aim is to make conclusions based on this information. More precisely, the functional dependencies can be determined in M * and the goal is to say something about the functional dependencies of the completely unknown M . There can be several interesting relationships between the two systems of functional dependencies, but in the present paper we investigate only the relationship between the sizes. (This will be later defined more precisely.). First consider Model 1. Suppose for instance that A → a (A ⊂ Ω, a ∈ Ω) holds in M . Then the data in a row in the columns of A determine the data of the same row in the column a. We know however only the corresponding rows in M * . The data in the columns of A in M * do not necessarily determine the data in a, since these data may be distorted. Can we enlarge A into an A whose data in M * already determine a? If yes, to what extent should it be enlarged?. For instance, if the number of errors in one row is at most one (e = 1), and in our previous example sex is one of the attributes then either the first name or the sex is correct. Yet, the definitely erroneous record (Maria, Sklodowska, M ) could be found in two different rows of M * (one obtained from (Mario, Sklodowska, M ) of M by changing the first name and the other one obtained from (Maria, Sklodowska, F of M by changing the sex). Further attributes might be needed to identify the individual. That is, (first name, last name, sex) is not a 1-error-correcting key (see the definition below).. Formalize our notions. The number of different entries of two sequences of the same length (two rows of the matrix) is the Hamming distance. If r, s are two such rows, this is denoted by h(r, s). If r = (r 1 , . . . , r n ) is a row of the matrix M , A = {i 1 , . . . , i α } is a set of columns of M then let r(A) denote the subrow (subsequence) of r determined by these columns: r(A) = (r i 1 , . . . , r iα ). If r is a row of M , let r * denote the corresponding row in M * . The only information we know about r * is that h(r, r * ) ≤ e.. Let C and B be two sets of columns of M . We write C → {e}B if the knowledge of r(C) for some row(s) r of M uniquely determines the subrow r(B) even after changing at most e values in the row(s) in any arbitrary way. More formally, we have C → {e}B iff the following holds for every pair r, s of rows of M : for all distorted versions r * and s * satisfying h(r, r * ) ≤ e, h(s, s * ) ≤ e the equation r * (C) = s * (C) implies r(B) = s(B). Observe that this is not necessarily true for the distorted versions r * (B) and s * (B), but they might differ only in at most 2e places. C → {e}B is called an e-error-correcting functional dependency. In case of C → {e}Ω we say that C is an e-errorcorrecting key.. Note that we actually formulated the e-error-correcting functional dependencies in terms of M . The name might be misleading: the transmitted information is not (even theoretically cannot be) corrected here. Only the functional dependency is "corrected" in the sense that it is recognizable even in M * in the following sense: if a certain functional dependency A → B holds in M , but does not hold in M * , A is enlarged ("corrected") into C to make the functional dependency "valid" even in M * .. The aim of the present paper is to find inequalities between the sizes of the sets occurring in the really existing functional dependencies in M and sizes of the sets occurring in the e-error-correcting ones. Our two conference papers [5] , [6] contain our results in a preliminary form, [5] deals with the case of the keys.. It is worth mentioning that {a} → {1}{a} does not hold, since the knowledge of the data in the column a does not give any information, it can be erroneous. It does not determine the value in column a.. The m × |C| submatrix of M determined by (corresponding to) the set C of its columns is denoted by M (C).. is an e-error-correcting functional dependency iff the pairwise Hamming distance of the rows r(C), s(C) of M (C) is at least 2e + 1 whenever r(B) = s(B).. Proof. First suppose that the Hamming distance of every pair of rows r(C), s(C) of M (C) is at least 2e+1 if the rows are different in some column a ∈ B (that is r({a}) = s({a})). In other words r(B) = s(B) implies h(r(C), s(C)) ≥ 2e + 1. Hence r * (C) = s * (C) follows. Then C → {e}B is an e-error-correcting functional dependency.. The converse is also true. Suppose that C → {e}B is an e-error-correcting functional dependency, and r(B) = s(B) holds for a pair of rows. Then h(r(C), s(C)) ≥ 2e + 1 must hold, otherwise one could find appropriate distorted versions r * (C) = s * (C), contradicting the definition. 2. Observe that the conditions are totally formulated in terms of M . Proposition 1.1 provides an alternative definition for e-error-correcting functional dependencies and hence we will use this form in the rest of the paper rather than the original definition. Model 1 is studied in Section 2.. Model 2 is motivated by data mining. Then only M * is known, nothing is known about M . In Model 1 the keys, functional dependencies in M are known for the receiver. We only want to derive conditions for the functional dependencies of M * from those of M . In case of data mining (Model 2) we have no prior information on M . Only M * is known, it defines some (virtual) functional dependencies. The question here is what the relationship between (the unknown) original dependencies (in M ) and the virtual ones (in M * ) is. Model 2 will be studied in Section 3.

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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 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 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