Integrality ratio for group Steiner trees and directed steiner trees We present an ~(log 2 k) lower bound on the integrality ratio of the flow-based relaxation for the Group Steiner Tree problem, where k denotes the number of groups; this holds even for input graphs that are Hierarchically WellSeparated Trees, introduced by Bartal [Symp. Foundations of Computer Science, pp. 184-193, 1996], in which case this lower bound is tight. This relaxation appears to be the only one that have been studied for the problem, as well as for its generalization, the Directed Steiner Tree problem. For the latter problem, our results imply an ~[ ~ ) integrality ratio, where n is the number of vertices in the graph. For both problems, this is the first known lower bound on the integrality ratio that is superlogarithmic in the input size. We also show algorithmically that the integrality ratio for Group Steiner Tree is much better for certain families of instances, which helps pinpoint the types of instances that appear to be most difficult to approximate. Group Steiner Tree is a network design problem that generalizes both Set Cover and the Steiner Tree problem. The Directed Steiner Tree problem is a further generalization of Group Steiner Tree. The polynomialtime approximability of these NP-hard problems is not yet understood. In particular, there is an intriguing gap between algorithms that achieve polylogarithmic approximation ratio (in quasi-polynomial time for the latter problem) and a logarithmic hardness of approximation that immediately follows from Set Cover. The only known relaxation for these problems is a natural flow-based linear programming relaxation. We show a polylogarithmic lower bound on the integrality ratio of this relaxation; this is the first such lower bound that is superlogarithmic in the input size. In fact, our bound is nearly tight in the important special case of input graphs which are tree networks. We also present improved approximation algorithms for certain families of instances of the Group Steiner Tree problem, shedding light on the type of instances that appear to be most difficult for the flow-based relaxation.. Our work unravels a major obstacle for achieving a logarithmic approximation ratio for these problems. We thus hope that it will lead to better approximation algorithms (say by an appropriate strengthening of the relaxation), or alternatively, to improved hardness results. We note that there is a (roughly) similar gap in many other optimization problems; several of these problems (e.g., bandwidth and cutwidth) have relaxations whose integrality ratio is at most polylogarithmic (see e.g. [DV01, BV02] ), but no superlogarithmic lower bound is known for the integrality ratios of these relaxations.. (a). The Group Steiner Tree problem. The (undirected) Group Steiner Tree problem is the following. Given an undirected graph G = (V, E), a collection of subsets (called groups) gz, g2,.-., gk of V, and a weight we _> 0 for each edge e C E, the problem is to construct a minimum-weight tree in G that spans at least one vertex from each group gi. We can assume without loss of generality that there is a distinguished vertex r E V (called the root) that must be included in the output tree. The case where [gil = 1 for all i is just the classical Steiner Tree problem; the case where G is a tree (or even a star) can be used to model the set cover prob.-lem. A natural flow-based relaxation for this problem is the following. Come up with a capacity Xe E [0, 1] for each edge e E E so that the capacities can support one unit of flow from r to gi, separately for each gi (as opposed to supporting a unit flow simultaneously for all gi). Subject to this constraint, we want to minimize ~e wex~. It is easy to check that the feasible solutions which satisfy x~ E {0, 1} for all e, exactly correspond to feasible solutions for the Group Steiner Tree prob--lem; hence, the above flow-based relaxation is indeed a valid linear programming (LP) relaxation for the problem. This is the, only known relaxation for this problem (and for some of its generalizations), and is the main subject of investigation in this paper.. We start with a useful definition from [Bar96] . (Item (ii) is slig:htly stronger than the original definition from [Bar96] , but can be assumed without loss of generality due to the analyses of [Bar96, Bar98, The first polylogarithmic approximation algorithm for the Group Steiner Tree problem was achieved in the elegant work of [GKR00] .. A brief sketch of their O(log n log log n log N log k)-approximation algorithm, where n = IVI and N = maxi [gil, is as follows. First, the powerful results of [Bar98] are used to appropriately reduce the problem to the case where G is a tree T, with an O(log n log log n) factor loss in the approximation ratio. T can be furthermore assumed to be a c-HST for any desired constant c > 1. Next, solve the flow-based LP relaxation on T and round the fractional solution into an integral solution for T by applying a novel randomized rounding approach that is developed in [GKR00] . It is established in [GKR00] that for any tree T, this randomized rounding leads to an O(log N log k)-approximation. Thus, for the input graph G, we get an O(log n log log n log N log k)-approximation. The work of [GKR00] has been extended and expanded in several ways: Their algorithm was derandomized in [CCGG98, Sri01] ; an alternative (combinatorial) algorithm is devised in [CEK02] ; the loss incurred by the reduction to an HST is improved to O(log n log log log n) in IBM03]. (We will discuss the Directed Steiner Tree problem below, but just mention for now that the same flow-based relaxation has been shown to have an integrality ratio of gl(x/k) for this problem [ZK02] .). Since the first appearance of a polylogarithmic approximation for the Group Steiner Tree problem (in the conference version of [GKR00] in 1998), there has been much interest in whether the approximation ratio can be improved. One concrete notable question in this regard has been the following: Can we achieve an approximation ratio better than O(log N log k) for trees? This is interesting for at least two reasons. First, since [GKR00] shows a reduction to the case of trees as seen above, an improved approximation for trees (or even for the case of c-HSTs for some constant c > 1) would directly lead to an improved approximation for general graphs. Further, even the case where G is a star (which is a tree) captures the Set Cover problem for which o(log k)-approximation is hard [Fei98] , so there is an intriguing gap even on trees.. Our main technical result is that for any constant c > 1, the integrality ratio of the flow-based relaxation for c-HSTs is ~(log 2 k). This bound is in fact tight, since an O(log 2 k) bound on the integrality ratio holds for c-HSTs; this is an unpublished work, resulting from our discussions with Anupam Gupta and R. Ravi. Recall that the upper bound of [GKR00] for trees in general is O(logNlogk); our methods show an fl(log N log k/log log N) lower bound on the integrality ratio, even for a class of HSTs. Such log-squared lower bounds had been conjectured by Uri Feige circa 1998. Our integrality ratio lower bound is shown via a random construction. The analysis is somewhat intricate, and requires delving into lower-order terms. We also show that the same lower bound holds also for trees where all weights are the same (i.e., unit-weight trees). Finally, we show randomized rounding algorithms for the flowbased relaxation that lead to improved approximation algorithms for certain special families of HSTs; this sheds light on the type of instances that are most difficult to approximate.. (b). The Directed Steiner Tree problem. This is the directed version of the (undirected) Steiner Tree problem. Given an edge-weighted directed graph that specifies a root vertex r and k terminal nodes Vl, v2,. • •, vk, the goal is to construct a minimum-weight out-branching tree rooted at r, which spans all the terminals vi. This problem is easily seen to generalize the undirected Group Steiner Tree problem, as well as to be equivalent to the directed Group Steiner Tree problem. Aside of intrinsic interest, this problem is also of current interest, e.g., in the context of multicasting in the Internet (where inter-node distances are often not symmetric). The polynomial-time approximation ratio currently known for this problem is k ~, for any constant > 0 [CCC+99] ; their algorithm extends to a polylogarithmic approximation ratio in quasi-polynomial running time. The flow-based relaxation here is similar: install a capacity x~ E [0, 1] so that a unit of flow can be shipped from r to vi, separately for any given i. Intriguingly, it was recently shown in [ZK02] that this relaxation has an integrality ratio of ~(~/k), precluding a polylog(k)-approximation algorithm based on this relaxation. However, the examples constructed in [ZK02] have k = O(Tr~s~; hence, the result of (log log n) fl [ZK02] does not imply an w(logn) integrality ratio. In summary, this work develops improved/tight lower bounds on the integrality ratio of the only known relaxation for Group Steiner Tree and Directed Steiner Tree; we also prove algorithmically that the integrality ratio for Group Steiner Tree is much better for certain families of instances, pinpointing the type of instances that appear difficult for this relaxation. Our hope is that this will spur new approaches/relaxations for the problem, or alternatively help us determine the limits to its polynomial-time approximability. Our results improve the current understanding of the integrality ratio of the flow-based relaxation for the Group Steiner Tree problem, but some very intriguing gaps still remain. Although for HSTs our ~2(log 2 k) lower bound is tight, for general trees there is a slight slackness between our ~(log k log N/log log N) lower bound and the O(log k log N) upper bound of [GKR00] . Interestingly, an O(log2(kN)/log log(kN))-approximation by a quasi-polynomial time algorithm is devised in [CEK02] ; their algorithm is combinatorial (i.e., not LP-based). Does their algorithm hint that the known upper bound on the integrality ratio in trees is not tight? Or maybe it hints that there is a separation between polynomial and quasi-polynomial (approximation) algorithms? A possible step towards closing this gap (in the integrality ratio on trees) is to analyze the following instance suggested by Uri Feige circa 1998: Take a complete tree of arity 4 (i.e., every non-leaf vertex has 4 children) and height log s k; now generate k groups, each containing k leaves, by an independent randomized branching process that starts from the root and picks two out of four children until the leaves are reached.. For general graphs, there is an even bigger slackness, as the known upper bound is 0(log n log k log N) [GKR00] and the lower bound is just the lower bound for trees described above. It is worth noting that a significantly better upper bound can be achieved in (general) graphs of small diameter. In particular, an O(log k) up--per bound for expander graphs is shown in IBM03]; this bound is tight since expanders contain a large star metric. We therefore set forth the following question, which was formulated together with Yair Bartal: What is the integrality ratio for the Group Steiner Tree problem on a (say two-dimensional) grid?. The shortest-path metric of a grid contains, up to constant distortion, an HST which is a complete regular tree (see e.g. [BBM01] ). This tree is similar to our tree 'Fn (and to Feige's tree described above), but differs in parameters like arity and weight; thus, one may suspect that the integrality ratio in grids is at least as large as in HSTs. In comparison, the best upper bound that we are aware of for two-dimensional grids is O(log n log k log N), by employing the [GKR00] approach with a specialized reduction of the grid to HST (using e.g. [KRS01]).

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


Detection of event-related potentials in individual subjects using support vector machines Abstract Event-related potentials (ERPs) are tiny electrical brain responses in the human electroencephalogram that are typically not detectable until they are isolated by a process of signal averaging. Owing to the extremely small size of ERP components (ranging from less than 1 lV to tens of lV), compared to background brain rhythms, statistical analyses of ERPs are predominantly carried out in groups of subjects. This limitation is a barrier to the translation of ERP-based neuroscience to applications such as medical diagnostics. We show here that support vector machines (SVMs) are a useful method to detect ERP components in individual subjects with a small set of electrodes and a small number of trials for a mismatch negativity (MMN) ERP component. Such a reduced experiment setup is important for clinical applications. One hundred healthy individuals were presented with an auditory pattern containing pattern-violating deviants to evoke the MMN. Two-class SVMs were then trained to classify averaged ERP waveforms in response to the standard tone (tones that match the pattern) and deviant tone stimuli (tones that violate the pattern). The influence of kernel type, number of epochs, electrode selection, and temporal window size in the averaged waveform were explored. When using all electrodes, averages of all available epochs, and a temporal window from 0 to 900-ms post-stimulus, a linear SVM achieved 94.5 % accuracy. Further analyses using SVMs trained with narrower, sliding temporal windows confirmed the sensitivity of the SVM to data in the latency range associated with the MMN. Event-related potential (ERP) components are specific deflections in the averaged waveform, which are considered to be tied to discrete stages of neural processing. They are extracted from trials or epochs of electroencephalogram (EEG) data that are temporally locked to a repeated event, either a stimulus or a behavioural response, which are segmented from the continuous EEG recording and then averaged together. This averaging process reduces the amplitude of brain activity that is unrelated to the event, thus presumed to be random with respect to it in time, and retains brain responses that are temporally related to the event (see [1] for a full description of ERP derivation). The most commonly used definition of an ERP component was outlined by Näätänen and Picton in 1987 [2] . Their definition emphasizes that an ERP component is described not only by the brain regions that contribute to its production, but also by the experimental parameters which must be satisfied in order to observe the response. ERPs can be evoked by stimulation in any sensory modality and can also occur in response to motoric and cognitive events.. In the auditory domain, ERPs can be used to index processing all the way from early sensory processing in the brainstem to cortical language processing. The auditory brainstem response (ABR) has a long history of use in both neurology and audiology [3] . Cortical ERPs have yet to become a routine part of clinical practice, even in audiology (reviewed in [4] ) and cognitive assessment of higher functions like language [5] [6] [7] where their potential utility is very clear.. In large part, the failure of cortical ERPs to be translated into clinical practice is due to the requirements of the signal-averaging process. Many repetitions of the evoking stimulus or event must be presented in order to isolate ERPs, which can lead to long testing times. Another important consideration for the application of ERPs for both clinical and research purposes is the need to statistically verify the presence of the response. Given the low signal-to-noise ratio of these responses versus background EEG, the typical approach in ERP research is to perform univariate statistics across groups, or across experimental conditions, using amplitudes that are measured in the averaged waveform [8] .. Many groups have proposed solutions to verify the presence of ERPs in individual subjects, and even single epochs, by applying techniques such as wavelet analysis [9] [10] [11] , independent component analysis [12] , integrated waveforms [13] , and nonparametric analyses [14] . In some very specific clinical situations, the approach of repeating stimulation until a statistical threshold is reached [15] , or using basic t-scores to evaluate the presence or the absence of a particular component [16, 17] , has been examined.. In particular, the field of Brain-Machine Interfaces (BMIs) has yielded much progress in the single-subject detection of responses like ERPs. This area seeks to use brain imaging data, mostly noninvasive technology, such as EEG, to identify brain waves in continuous EEG recordings that can be interpreted as commands by a computer [18] . The progress that has been seen in this area is mainly due to advances in machine learning techniques. While results in BMI research show that signals can be extracted from single trials, here we investigate specifically the use of machine learning techniques for the detection of ERPs in individual subjects for the purpose of clinical evaluations. Support vector machines (SVMs) are evaluated for their applicability as part of the protocol known as the Halifax Consciousness Scanner [19] . Our approach is hence mediating between a classical clinical approach and methods of BMIs.. The SVMs are a form of supervised statistical machine learning (for a primer, see [20] ) and were first described by Vapnik and colleagues in 1992 [21] . When classified cases are used to train an SVM, it derives an optimal function based on the features that separate the two classes. Unlike other statistical classification methods, such as discriminant function analysis and linear regression, SVMs derive their decision function from only a subset of the data-the cases that are difficult to classify. These cases are referred to as support vectors. The goal of an SVM is to maximize the distance between the decision function's boundary and the support vectors. They can operate in either a linear or a nonlinear fashion and are notable among machine learning techniques in view of their ability to produce generalizable models from small datasets [22] .. The SVMs have demonstrated very high accuracy for discriminating between experimental conditions [23] , and moderately high accuracy for the prediction of clinical group membership [8] based on ERP features. The present work arose from the development of a novel method for evaluating brain function using ERPs, known as the Halifax Consciousness Scanner (HCS; described in [19] ). The initial intent of the HCS is for applications in brain-injured populations. This context presents very different challenges from BMIs because patients may be unable to comply with task instructions [24] . It therefore uses stimulation parameters that do not require effortful task completion. This approach is also beneficial for clinical populations that may seek to influence their test results to conceal, augment, or even fake an injury [25] . Yet another important feature that distinguishes clinical applications from typical SVM applications in BMIs is the need for rapid ERP detection, without any kind of training period for the classifier. For this reason, we apply SVM training across subjects, rather than in the within-subject fashion that is typical for BMIs.. To evaluate the ability of an SVM to verify ERPs in individual subjects, the present study uses an ERP known as the mismatch negativity (MMN). First discovered by Näätänen and colleagues in 1978, the auditory MMN is an ERP that indexes the detection of change in a sequence of sounds [26] . The MMN is linked to a level of auditory memory that is believed to form the basis of conscious perception [27] , and is an important part of the HCS method [19] , particularly due to its well-established utility for coma monitoring [15, [28] [29] [30] [31] [32] . Owing to its small size, the MMN presents a challenge for detection at the individual subject level.. The MMN is commonly evoked using an experimental design known as an oddball paradigm. Although it is well understood that the oddball paradigm frequently causes an undesirable overlap of other ERPs with the MMN [33] , it provides a simple framework for understanding how the MMN is obtained. In the oddball paradigm, a sequence of identical ''standard'' stimuli is interrupted at random and unpredictable times by ''deviant'' stimuli that possess some changed features, such as tonal frequency, intensity, or duration. The MMN is observed as an enhancement in negative voltage that occurs approximately 100-250 ms from deviant stimulus onset. It rests atop other ERP components that are associated with sensory processing, such as the ''obligatory'' P1-N1-P2 complex, and is isolated from these responses through a process of subtraction in which the averaged waveform to the standard stimuli is subtracted point-by-point from the averaged waveform to the deviant stimuli. This produces the so-called ''difference wave,'' in which the MMN appears as a negative deflection in the 100-250 ms range. Although the amplitude of the MMN can be enhanced by experimental parameters such as reducing the silent period between stimuli (the interstimulus interval, or ISI), or increasing the difference between the standard and deviant tones [34] , it is often quite small in amplitude, being around 1 lV.. The MMN is not restricted to simple oddball deviance and can also be evoked by more complex violations of concrete or abstract rules [35] . Under these circumstances, a pure MMN should be obtained, which is free of overlap from other ERP components. In the present study, the MMN is evoked in response to violations of a simple pattern of two alternating tones. Two types of SVM analyses are performed: one to achieve high classification accuracy (SVM acc ), and the other to help one clarify whether the waveform features that are associated with the MMN are indeed used by the SVM (SVM time ). The SVM acc analysis explores a variety of factors which could impact classifier accuracy: number of epochs entered into the averaged waveforms, scalp electrode selection, kernel type, and temporal window selection.. It is generally accepted that, as the number of epochs entered into the ERP waveform increases, the SNR of the ERP improves versus background EEG [1] . For the MMN, the number of epochs that is generally recommended for averaging is at least 150 [36] , although robust MMNs have been observed with approximately 60 [34] . For this reason, the first SVM analysis explores the influence of number of epochs entered into the average on the accuracy of the classifier.. The typical scalp distribution for the MMN is maximal at frontal sites, and inverted at sites below the Sylvian fissure when a nose reference is used. This distribution has been attributed to bilateral, vertically oriented cortical generators on the supratemporal plane, as well as a frontal source [37] . Thus, a classifier that is limited to frontal sites might achieve the highest performance. However, for clinical populations such as patients with acquired brain injuries, atypical scalp distributions are expected [38] . For this reason, the value of using scalp electrode sites separately, or combined, is evaluated.. The so-called kernel trick permits SVMs to operate in a nonlinear fashion by projecting low-dimensional data into higher-dimensional space [20] . This may provide a better fit to complex datasets, but this improved accuracy may come at the cost of overfitting-a lack of generalizability of the results to new datasets [20] . For this reason, a linear classifier may be preferable, but the values of different kernel types are examined to determine whether significant improvements in performance can be achieved over the linear solution.. The MMN is expected to occur approximately 100-250 ms from stimulus onset [36] ; however, other information in the ERP waveforms may be useful for classification. The portion of the waveform that is entered into the feature vectors is therefore varied, to determine what temporal window produces the maximum performance. A difficulty in the interpretation of SVMs, however, is that they derive their rules using a complex combination of factors. Any waveform features that are within the selected temporal window, be they part of the MMN or not, can contribute to the decision function. Thus, the SVM time analysis is performed, not to achieve high accuracy, but to provide finer detail regarding which temporal regions of the waveforms contain detectable information for classification. The tone sequence evoked a moderate-sized, frontally maximal MMN component that inverted in polarity at the mastoid site. This scalp distribution, though observed on a reduced array of sensors, conforms with the classic 3 Grand average waveforms for standard tones (black), repetition deviants (gray), and the deviant minus standard difference wave (dotted). All amplitudes are in microvolts and all time values are in milliseconds. A sharp hardware-related deflection was observed at stimulus onsets (i.e., at 0 ms and every 265 ms thereafter). The MMN is indicated at the Frontal and Mastoid sites by an arrow. In the standard stimulus waveforms, obligatory auditory responses including P1, N1, and P2 were apparent. Because of the rapid rate of stimulus delivery, these took on the appearance of a steady state response Detecting ERPs using SVMs 7 morphology of the MMN and is identical to that observed using similar pattern paradigms [40] . The SVM acc analysis demonstrated that standard tone waveforms and deviant tone waveforms could be classified with 94.5 % accuracy (as shown in Fig. 6 for the all sites feature vector at window size 900 ms). These results bode well for the use of SVMs to detect ERPs for clinical applications, such as the ERP test battery approach that has been developed by our group, known as the Halifax Consciousness Scanner (described in [19] , and a companion paper, [14] .. The performance of the SVM classifier was affected by a number of factors. First, the number of epochs used in the standard and deviant tone averaged waveforms was explored using a linear SVM. The accuracy of the classifier increased directly as a function of the number of epochs entered into the average. This result is entirely consistent with the general belief that the averaging process increases the SNRs of ERPs [1] . There was a consistent trend in which accuracy increased as the number of epochs entered into the average was increased from 2 to 40. The effect of Random Trials Fig. 4 Accuracy of a linear SVM as a function of the number of standard and deviant tone epochs entered into the average (using all scalp sites). Accuracy increased as the number of epochs entered into the average was increased from 2 to 40, but increased much more steeply in the serial-than the random-selection condition, and reached a higher maximum when 40 epochs were used (78.13 %, SD = 0.02 % for serial selection, versus 65 %, SD = 0.043 % for random selection). number of epochs on the accuracy of the SVM was much more dramatic for serially than randomly selected epochs, and achieved a higher accuracy when using 40 epochs in the serial condition. These results suggest that sequential effects that emerge over the stimulation protocol may have contributed to the accuracy of the classifier.. The effect of averaging up to 67 epochs (the full available set) could not be explored across the group, due to the availability of less than 67 deviant epochs in some subjects as a result of artifact rejection. Nevertheless, the full set of available epochs was used for each subject in the final SVM solution, which achieved an accuracy of 94.5 %. Although the recommended number of epochs for MMN derivation is 150, with a typical recording time of 6-12 min [36] , the very high accuracy that was observed here with only 67 deviants, presented in an optimized 2.5-min sequence, is promising for clinical applications. While optimal recording conditions may be difficult to achieve, particularly in a clinical environment, the relatively small amount of data required for classification using this method represents a significant reduction in testing time versus conventional stimulation parameters. Short testing times are necessary in a clinical context because they interfere minimally with routine care procedures and avoid fluctuations in vigilance, attention, and fatigue which can cause false negatives on ERP-based tests [42] .. Other factors that influenced SVM accuracy were the type of kernel (linear, RBF, cubic, or quadratic), the temporal window used, and the electrode site used. When all kernel types were explored across different temporal windows, using a feature vector with data from all sites, the linear SVM clearly outperformed other kernel types. This effect was particularly noticeable as the temporal window size approached 200 ms-the period in which the MMN begins to emerge. At the maximum window size, 900 ms, Detecting ERPs using SVMs 9 the linear SVM performed significantly better than all other kernel types, particularly the RBF, which never rose above 59 %. It remains possible that optimizing the parameters of each SVM kernel type could improve performance for the nonlinear classifiers. More complex kernel types, however, can lead to overfitting, and a lack of generalizability [20] . For this reason, the effect of parameter optimization was not explored for the nonlinear classifiers. The electrode site that was entered into the feature vector also had a significant impact on the accuracy of the classifier. In line with the known scalp distribution of the MMN [37] , feature vectors from the Frontal and Central site produced significantly better results than ones from the Parietal site. Similarly, feature vectors containing data from all three scalp sites produced very high performance, which was not significantly different from the classifiers that were trained exclusively with data from the Frontal and Central sites. This approach may be preferable in clinical populations whose scalp distributions can vary from that typically observed in healthy individuals.. For the linear SVM, increasing the width of the temporal window entered into the feature vector had a direct impact on the accuracy of the classifier, in which increasing the width of the window systematically improved performance. The most dramatic improvement in accuracy occurred around 200 ms, near the peak of the MMN in the grand average (193 ms). Maximum accuracy was observed when the full post-stimulus epoch was utilized (900 ms).. One drawback of SVMs for classification, as opposed to other statistical classifiers like discriminant function analysis, is that their decision functions use complex combinations of information that can make their interpretation difficult. Thus, it is unknown which ERPs or brain oscillations that lie in the employed temporal window are contributing to the performance of the classifier.. To gain insight into the waveform features that contribute most to classification accuracy, a secondary analysis (SVM time ) was performed. Of the 46 SVMs trained in nonoverlapping 20-ms (5 sample) windows, those in the 160-260 and 720-820 ms range achieved the highest accuracy (first period maximum = 86.5 %, SD = 0.1 %, second period maximum = 83.5 %, SD = 0.058 %). The first period corresponds well with the expected latency of the MMN to the pattern deviant, which peaked on average at 193 ms. The second period corresponds with a region of positivity in the deviant tone response, which could reflect a P3-type response. While both MMN and P3 indicate that deviance detection has occurred, P3 is linked to subsequent cognitive operations like attention switching and working memory updating [43] . Regardless of the true component structure of this positive response, it is clear that a classifier which uses this portion of the waveform does not detect the MMN per se, but rather uses a larger variety of waveform features to perform the task of classifying waveforms associated with deviance detection.. In conclusion, linear SVMs proved to be effective at classifying the MMN. This method could be extremely valuable for clinical applications of the MMN, and other ERPs, by both reducing testing time versus conventional stimulation parameters and permitting the application of ERPs at the single subject level. Among the potential applications for the MMN is monitoring the status of severely brain-injured patients [15, 19, 28, 32] . Further development for use in this patient population may require additional testing to determine efficacy in the context of the severely altered EEG patterns that are commonly observed in this group [38] . Results must also be interpreted with care when applied in the absence of more concrete indicators of neurological status [44] [45] [46] [47] .

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


Automatic

generation

of

outdoor

and

indoor

building

scenes

from

a

single

image

Abstract

In

this

paper,

a

novel

approach

for

creating

models

of

building

scenes

is

presented.

The

proposed

method

is

fully

automated

and

fast,

and

accurately

reconstructs

both

outdoor

images

of

a

building

and

indoor

scenes,

with

perspective

cues

in

real-time,

using

only

one

image.

It

combines

the

extracted

line

segments

to

identify

the

vanishing

points

of

the

image,

the

orientation,

the

different

planes

that

are

depicted

in

the

image

and

concludes

whether

the

image

depicts

indoor

or

outdoor

scenes.

In

addition,

the

proposed

method

efficiently

eliminates

the

perspective

distortion

and

produces

an

accurate

model

of

the

scene

without

any

intervention

from

the

user.

The

main

innovation

of

the

method

is

that

it

uses

only

one

image

for

the

reconstruction,

while

other

state-of-the-art

methods

rely

on

the

processing

of

multiple

images.

A

website

and

a

database

of

images

were

created

to

prove

the

efficiency

of

the

proposed

method

in

terms

of

time

needed

for

the

reconstruction,

its

automation

and

model

accuracy

and

can

be

used

by

anyone

so

as

to

easily

produce

user-generated

content:

One

of

the

most

challenging

topics

in

Computer

Vision

is

to

identify

the

way

that

computers

can

perceive

objects

from

images.

The

process

to

recover

the

information

of

an

object

from

images

is

called

reconstruction.

When

a

human

looks

at

a

picture

is

really

easy

for

him/her

to

understand

how

the

scene

is

constructed

by

using

his/her

experience

of

the

man-made

world.

On

the

other

hand,

computers

and

robots

do

not

have

such

ability.

They

can

only

see

the

pixels

of

the

image.

Even

when

more

intelligent

algorithms

are

used

they

can

find

lines

and

planes

but

they

are

incapable

of

combining

all

these

contents

together

and

form

a

structure.

As

a

scientific

discipline,

computer

vision

is

concerned

to

deal

with

the

theory

of

building

artificial

systems

that

obtain

information

from

images.

State-of-the-art

methods

that

have

been

reported

in

the

literature

and

use

more

than

one

images

for

achieving

reconstruction

follow

completely

different

concept

from

the

proposed

algorithm.

These

algorithms

do

not

depend

on

criteria

such

as

lines

and

planes

extraction.

In

order

to

reconstruct

a

scene

they

take

advantage

of

the

position

of

specific

points

in

the

different

views

of

the

scene.

Consequently,

having

multiple

images

of

a

scene

the

need

to

make

assumptions

about

the

construction

of

the

scene

is

reduced

or

eliminated.

Thus,

these

methods

could

reconstruct

greater

variety

of

pictures

however

they

face

two

main

disadvantages:

firstly,

they

require

greater

amount

of

input

data

to

perform

the

reconstruction,

which

are

not

always

available,

especially

when

dealing

with

real

life

cases

and

secondly,

the

time

required

for

such

a

reconstruction

is

higher

when

compared

to

the

real-time

method

proposed

in

this

paper..

reconstruction

from

a

single

image

is

definitely

a

challenging

problem

and

many

researchers

have

dealt

with

it.

The

purpose

of

the

reconstruction

from

a

single

image

is

what

humans

can

do

to

perceive

information.

Both

automated

methods

and

methods

that

need

user's

interaction

exist.

The

most

well

known

commercial

method,

which

is

based

on

user

interaction,

is

Google

Sketch-Up

Other

algorithms

that

work

with

the

assistance

from

the

users

are

Façade

Tour

into

the

Picture

Single

View

Metrology

and

ATIP.

This

paper

focuses

on

fully

automated

methods,

which

can

be

classified

into

three

main

categories

depending

on

different

assumptions:

depth

reconstruction,

geometrical

projection

and

image

content-based

methods.

The

methods

based

on

depth

analyze

the

distance

between

each

pixel

and

the

camera,

and

the

methods

based

on

the

image

content

try

to

find

the

similar

area

and

combine

the

pixels

which

belong

to

the

same

object.

However,

simple

observation

cannot

show

the

effectiveness

of

the

methods

to

solve

the

problem.

For

example,

some

methods

are

more

effective

for

urban

images

and

not

for

natural

scenes.

"Automatic

Photo

Pop-Up"

is

the

first

fully

automated

method

for

reconstruction.

The

method

is

based

on

training

data

and

works

for

outdoor

photos,

both

man-made

and

natural.

The

computational

time

of

this

method

for

an

image

is

about

min

using

unoptimized

MATLAB

code

on

a

GHz

Athalon

machine

The

main

drawbacks

of

this

algorithm

is

that

it

relies

on

statistics,

thus

wrong

models

may

be

produced

and

it

cannot

be

used

for

real-time

reconstructions.

One

of

the

most

well

known

algorithms

is

"Make

by

Saxena

et

al.

where

depth

estimation

from

a

single

still

image

is

automatically

estimated.

This

method

is

based

on

a

collection

of

a

training

set

of

monocular

images

of

unconstructed

indoor

and

outdoor

environments,

which

include

forests,

sidewalks,

trees,

buildings,

etc.

Then,

a

supervised

learning

technique

is

applied

to

predict

the

value

of

the

depthmap

of

the

image.

The

distinct

advantage

of

this

method

is

that

there

are

no

any

assumptions

for

any

specific

features.

However,

the

multiscale

Markov

Random

Field

(MRF)

model

used

is

complex

and

time

consuming

for

real-time

reconstruction

from

a

image.

Furthermore,

in

geometric

constraints

are

used

to

recover

an

indoor

structure

from

a

single

image.

More

specifically,

several

assumptions

are

generated

about

the

structure

of

the

image

by

joining

the

line

segments,

that

were

previous

extracted,

and

producing

angles

and

joining

angles

to

create

the

whole

structure.

An

evaluation

process

then

takes

place

in

order

to

select

the

best

representation

of

the

indoor

scene

from

the

previous

assumptions

and

the

model

reconstruction

follows.

The

algorithm

produces

good

results

even

if

the

scene

has

many

occluding

objects

but

is

functional

only

in

indoor

images.

In

the

same

line

with

this

work,

deals

with

the

problem

of

reconstruction

of

indoor

scenes

with

perspective

cues.

With

this

method

an

accurate

model

is

produced

but

without

resolving

the

problem

of

projective

distortion

and

without

using

texture

mapping

at

the

models.

Another

recent

algorithm

is

presented

in

where

a

dynamic

Bayesian

network

model

capable

of

recovering

information

from

many

images

is

presented.

The

model

assumes

a

"floorwall"

geometry

of

the

scene

and

is

trained

to

recognize

the

floor-wall

boundary

in

each

column

of

the

image.

The

method

depends

on

the

calibration

of

a

camera

and

the

parameters

of

the

camera

are

not

always

available..

In

this

paper,

geometry

is

derived

from

line

segments

in

one-point

perspective

indoor

image

that

consists

largely

of

orthogonal

planes,

and

from

the

exterior

of

buildings.

The

aim

of

this

work

is

to

present

a

novel

algorithm

for

real-time

scene

generation

from

a

single

image

able

to

be

used

for

both

indoor

and

outdoor

images

of

buildings,

to

be

fully

automated,

real-time,

without

any

user

interaction.

The

proposed

method

is

based

on

detecting

and

fitting

the

line

segments

correctly

so

as

to

form

the

structure

of

the

indoor

or

outdoor

building

scene.

In

addition,

it

deals

with

the

problem

of

perspective

distortion

and

by

correcting

it,

it

generates

accurate

models.

The

work

presented

in

this

paper

is

fully

aligned

with

the

scope

of

this

special

issue

since

it

provides

an

easy

to

use

tool

for

automatic

creation

of

user-generated

content

using

a

single

image

The

rest

of

this

paper

is

organized

as

follows:

In

Section

all

the

distinct

steps

of

the

proposed

algorithm

are

presented

in

detail

along

with

innovations

which

substantially

improve

the

current

state-of-the-art.

In

Section

the

relevant

website

is

shown.

Finally,

conclusions

are

drawn

in

Section

In

this

paper,

a

novel

algorithm

is

proposed,

able

to

create

a

reconstruction

of

a

building

scene

(indoor

and

outdoor),

fully

automated

without

any

user

intervention

from

a

single

image.

The

algorithm

is

free

to

be

tested

in

the

website

The

proposed

method

manages

to

create

models

in

a

fully

automated

manner.

Furthermore,

there

is

no

need

for

camera

calibration

and

the

external

and

internal

parameters

of

the

camera

are

not

necessary..

The

problem

that

is

attempted

to

be

solved

is

new

and

challenging

and

different

approaches

have

been

proposed

the

past

few

years.

Further

research

is

needed

in

order

to

reconstruct

all

the

possible

indoor

building

scenes

despite

the

presence

of

occluding

objects.

Moreover,

the

algorithm

may

be

further

extended

to

object

or

human

recognition

inside

a

scene,

so

as

to

fully

reconstruct

a

scene

using

only

a

single

image..

The

purpose

of

a

single

image

reconstruction

method

except

for

its

clear

research

challenges,

depicts

the

reality

more

accurately

since

by

using

such

a

method

a

building

can

be

reconstructed

with

its

real

dimensions.

Moreover,

models

are

widely

used

nowadays

on

the

Internet

either

for

navigation

or

for

visualization

and

presentation

of

monuments

and

other

cultural

heritage

sites

from

all

over

the

world.

For

example,

Google

Earth

uses

extensively

models

in

order

to

make

the

navigation

and

presentation

of

such

sites

more

fascinating.

The

single

image

reconstruction

algorithms

produce

models

that

have

small

size

and

can

be

distributed

easily

over

the

Internet.

Generally

speaking,

reconstruction

is

a

necessary

procedure

for

the

architects.

Nowadays,

most

of

the

projects

of

the

architects

are

presented

not

only

with

images

and

technical

drawings

but

also

with

models

in

order

to

capture

the

interest

of

the

client

and

to

visualize

the

buildings

with

more

details.

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


Bounded repairability of word languages What do you do if a computational object (e.g. program trace) fails a specification? An obvious approach is to perform a repair : modify the object minimally to get something that satisfies the constraints. This approach has been investigated in the database community, for integrity constraints, and in the AI community for propositional logics. Here we study how difficult it is to repair a document in the form of a string. Specifically, we consider number of edits that must be applied to an input string in order to satisfy a given target language. This number may be unbounded; our main contribution is to isolate the complexity of the bounded repair problem based on a characterization of the regular languages that admit bounded repairr. We consider the settings where the repair strategy is unconstrained and when the editing must be produced in a streaming way, i.e. by a letter-to-letter transducer. When a computational object does not satisfy a specification, an obvious approach is to repair it -edit it minimally so that it becomes valid. We may want to perform this editing transformation on the object, or we may be merely interested in knowing how difficult it would be to perform -that is, determining how far a given object or collection of objects is from satisfying the specification. In the database community, this has been extensively studied under the notion of constraint repair (see e.g. [1, 2] ): the specifications considered there are relational integrity constraints, such as keys and foreign keys, and the problems considered include determining how much a database needs to be modified in order to satisfy a given constraint.. Here we study repairs of words. In this case, we can simply consider the edit distance between strings, a standard measure of how many basic operations it takes to get from one string to another. Edit distance is lifted in a natural way to give a measure of the distance of a string u to a language (collection of strings) L. It is well known [3] that the standard dynamic programming approach to edit distance extends to give an efficient algorithm for calculating the minimum distance of u to any string in L, when L is a regular language given as a non-deterministic finite state automaton.. In this work we take the next step and consider a notion of "distance" between languages: given two languages R and T (specified in terms of automata), we aim to calculate how difficult it is to transform a string satisfying R into a string satisfying T . The notion is motivated by considering R to be a restriction -a constraint that the input is guaranteed to satisfy -while T is a target -a constraint that we want to enforce. We consider the worst-case over any string u ∈ R of the number of edit operations needed to move u into T , that is, we look at the worst-case number of edits needed to get from R to T . Of course, this number may be infinite; the core of our results is a procedure for solving the bounded repair problem, namely, to determine whether the supremum above is finite. In order to solve this effectively, we need to restrict the languages R and T . We consider this problem when the restriction and target languages are presented by both deterministic and non-deterministic finite state automata. In these cases we determine the complexity of the bounded repair problem.. Above we considered the use of an edit/correction function that can read the whole string in memory. In this work we consider the impact of limitations on the editing process -what happens when we require the editing to be done by a transducer, reading the input letter-by-letter and producing the corrected output, based only on a finite amount of control state and a fixed amount of lookahead in the word. We refer to this as a streaming repair processor. We isolate the complexity of the streaming repair problem for any lookahead and for any of type of representation of the languages considered in the non-streaming setting.. The above deals with the problem of determining whether the distance between two specifications is finite or infinite. But in the finite case, we may want to compute this distance exactly, and to produce the processor that optimally edits a given specification. Note that in the non-streaming setting, it is easy to describe the optimal processor: it is simply the procedure that given a word u runs a dynamic programming algorithm to compute the edit distance to the target language (e.g. the algorithm from [3] ). However, in the streaming setting it is not clear how to derive the optimal editing algorithm efficiently. We give results on the complexity of computing the exact bound when it is finite in both the streaming and non-streaming settings, and also give procedures for computing the optimal processor in the streaming setting.. The streaming and non-streaming repair problems have very different flavours: the former are closely related to games played on the components of two automata, while the latter require a more global analysis, and exhibit a close relation to distance automata. However, there are connections between the different problems: we show that in the case where there is no restriction, the bounded repair problems are the same for both the streaming and non-streaming setting. We also show that the bounded repair problem in the streaming setting is independent of the lookahead, and is robust under plausible alternative definitions.. In summary our contributions are:. 1.. We formalize the bounded repair problem for languages of words and characterize when regular languages have bounded repair, in both the streaming and non-streaming setting.

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


Computing good nash equilibria in graphical games This paper addresses the problem of fair equilibrium selection in graphical games. Our approach is based on the data structure called the best response policy, which was proposed by Kearns et al. [13] as a way to represent all Nash equilibria of a graphical game. In [9] , it was shown that the best response policy has polynomial size as long as the underlying graph is a path. In this paper, we show that if the underlying graph is a bounded-degree tree and the best response policy has polynomial size then there is an efficient algorithm which constructs a Nash equilibrium that guarantees certain payoffs to all participants. Another attractive solution concept is a Nash equilibrium that maximizes the social welfare. We show that, while exactly computing the latter is infeasible (we prove that solving this problem may involve algebraic numbers of an arbitrarily high degree), there exists an FPTAS for finding such an equilibrium as long as the best response policy has polynomial size. These two algorithms can be combined to produce Nash equilibria that satisfy various fairness criteria. In a large community of agents, an agent's behavior is not likely to have a direct effect on most other agents: rather, it is just the * Supported by the EPSRC research grants "Algorithmics of Network-sharing Games" and "Discontinuous Behaviour in the Complexity of randomized Algorithms". 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. EC'07, J une [11] [12] [13] [14] [15] 2007 , San D iego, California, USA. Copyright 2007 ACM 978-1-59593-653-0/07/0006 ...$5.00. agents who are close enough to him that will be affected. However, as these agents respond by adapting their behavior, more agents will feel the consequences and eventually the choices made by a single agent will propagate throughout the entire community. This is the intuition behind graphical games, which were introduced by Kearns, Littman and Singh in [13] as a compact representation scheme for games with many players. In an n-player graphical game, each player is associated with a vertex of an underlying graph G, and the payoffs of each player depend on his action as well as on the actions of his neighbors in the graph. If the maximum degree of G is Δ, and each player has two actions available to him, then the game can be represented using n2 Δ+1 numbers. In contrast, we need n2 n numbers to represent a general n-player 2-action game, which is only practical for small values of n. For graphical games with constant Δ, the size of the game is linear in n.. One of the most natural problems for a graphical game is that of finding a Nash equilibrium, the existence of which follows from Nash's celebrated theorem (as graphical games are just a special case of n-player games). The first attempt to tackle this problem was made in [13] , where the authors consider graphical games with two actions per player in which the underlying graph is a boundeddegree tree. They propose a generic algorithm for finding Nash equilibria that can be specialized in two ways: an exponential-time algorithm for finding an (exact) Nash equilibrium, and a fully polynomial time approximation scheme (FPTAS) for finding an approximation to a Nash equilibrium. For any > 0 this algorithm outputs an -Nash equilibrium, which is a strategy profile in which no player can improve his payoff by more than by unilaterally changing his strategy.. While -Nash equilibria are often easier to compute than exact Nash equilibria, this solution concept has several drawbacks. First, the players may be sensitive to a small loss in payoffs, so the strategy profile that is an -Nash equilibrium will not be stable. This will be the case even if there is only a small subset of players who are extremely price-sensitive, and for a large population of players it may be difficult to choose a value of that will satisfy everyone. Second, the strategy profiles that are close to being Nash equilibria may be much better with respect to the properties under consideration than exact Nash equilibria. Therefore, the (approximation to the) value of the best solution that corresponds to an -Nash equilibrium may not be indicative of what can be achieved under an exact Nash equilibrium. This is especially important if the purpose of the approximate solution is to provide a good benchmark for a system of selfish agents, as the benchmark implied by an -Nash equilibrium may be unrealistic. For these reasons, in this paper we focus on the problem of computing exact Nash equilibria.. Building on ideas of [14] , Elkind et al. [9] showed how to find an (exact) Nash equilibrium in polynomial time when the underlying graph has degree 2 (that is, when the graph is a collection of paths and cycles). By contrast, finding a Nash equilibrium in a general degree-bounded graph appears to be computationally intractable: it has been shown (see [5, 12, 7] ) to be complete for the complexity class PPAD. [9] extends this hardness result to the case in which the underlying graph has bounded pathwidth.. A graphical game may not have a unique Nash equilibrium, indeed it may have exponentially many. Moreover, some Nash equilibria are more desirable than others. Rather than having an algorithm which merely finds some Nash equilibrium, we would like to have algorithms for finding Nash equilibria with various sociallydesirable properties, such as maximizing overall payoff or distributing profit fairly.. A useful property of the data structure of [13] is that it simultaneously represents the set of all Nash equilibria of the underlying game. If this representation has polynomial size (as is the case for paths, as shown in [9] ), one may hope to extract from it a Nash equilibrium with the desired properties. In fact, in [13] the authors mention that this is indeed possible if one is interested in finding an (approximate) -Nash equilibrium. The goal of this paper is to extend this to exact Nash equilibria. We have studied the problem of equilibrium selection in graphical games on bounded-degree trees. We considered several criteria for selecting a Nash equilibrium, such as maximizing the social welfare, ensuring a lower bound on the expected payoff of each player, etc. First, we focused on the algebraic complexity of a social welfare-maximizing Nash equilibrium, and proved strong negative results for that problem. Namely, we showed that even for graphical games on paths, any algebraic number α ∈ [0, 1] may be the only strategy available to some player in all social welfaremaximizing Nash equilibria. This is in sharp contrast with the fact that graphical games on trees always possess a Nash equilibrium in which all players' strategies are rational numbers.. We then provided approximation algorithms for selecting Nash equilibria with special properties. While the problem of finding approximate Nash equilibria for various classes of games has received a lot of attention in recent years, most of the existing work aims to find -Nash equilibria that satisfy (or are -close to satisfying) certain properties. Our approach is different in that we insist on outputting an exact Nash equilibrium, which is -close to satisfying a given requirement. As argued in the introduction, there are several reasons to prefer a solution that constitutes an exact Nash equilibrium.. Our algorithms are fully polynomial time approximation schemes, i.e., their running time is polynomial in the inverse of the approximation parameter , though they may be pseudopolynomial with respect to the input size. Under mild restrictions on the inputs, they can be modified to be truly polynomial. This is the strongest positive result one can derive for a problem whose exact solutions may be hard to represent, as is the case for many of the problems considered here. While we prove our results for games on a path, they can be generalized to any tree for which the best response policies have compact representations as unions of rectangles. In the full version of the paper we describe our algorithms for the general case.. Further work in this vein could include extensions to the kinds of guarantees sought for Nash equilibria, such as guaranteeing total payoffs for subsets of players, selecting equilibria in which some players are receiving significantly higher payoffs than their peers, etc. At the moment however, it is perhaps more important to inves-tigate whether Nash equilibria of graphical games can be computed in a decentralized manner, in contrast to the algorithms we have introduced here.. It is natural to ask if our results or those of [9] can be generalized to games with three or more actions. However, it seems that this will make the analysis significantly more difficult. In particular, note that one can view the bounded payoff games as a very limited special case of games with three actions per player. Namely, given a two-action game with payoff bounds, consider a game in which each player Vi has a third action that guarantees him a payoff of Ti no matter what everyone else does. Then checking if there is a Nash equilibrium in which none of the players assigns a nonzero probability to his third action is equivalent to checking if there exists a Nash equilibrium that satisfies the payoff bounds in the original game, and Section 5.1 shows that finding an exact solution to this problem requires new ideas.. Alternatively it may be interesting to look for similar results in the context of correlated equilibria (CE), especially since the best CE may have higher value (total expected payoff) than the best NE. The ratio between these values is called the mediation value in [1] . It is known from [1] that the mediation value of 2-player, 2-action games with non-negative payoffs is at most 4 3 , and they exhibit a 3-player game for which it is infinite. Furthermore, a 2-player, 3action example from [1] also has infinite mediation value. Our approximation scheme (Theorem 3 and Theorem 4) shows a contrast between the games that we study and two-player n-action games, for which the corresponding problems are usually intractable. For two-player n-action games, the problem of finding Nash equilibria with special properties is typically NP-hard. In particular, this is the case for Nash equilibria that maximize the social welfare [11, 6] . Moreover, it is likely to be intractable even to approximate such equilibria. In particular, Chen, Deng and Teng [4] show that there exists some , inverse polynomial in n, for which computing an -Nash equilibrium in 2-player games with n actions per player is PPAD-complete.. Lipton and Markakis [15] study the algebraic properties of Nash equilibria, and point out that standard quantifier elimination algorithms can be used to solve them. Note that these algorithms are not polynomial-time in general. The games we study in this paper have polynomial-time computable Nash equilibria in which all mixed strategies are rational numbers, but an optimal Nash equilibrium may necessarily include mixed strategies with high algebraic degree.. A correlated equilibrium (CE) (introduced by Aumann [2]) is a distribution over vectors of players' actions with the property that if any player is told his own action (the value of his own component) from a vector generated by that distribution, then he cannot increase his expected payoff by changing his action. Any Nash equilibrium is a CE but the converse does not hold in general. In contrast with Nash equilibria, correlated equilibria can be found for low-degree graphical games (as well as other classes of conciselyrepresented multiplayer games) in polynomial time [17] . But, for graphical games it is NP-hard to find a correlated equilibrium that maximizes total payoff [18] . However, the NP-hardness results apply to more general games than the one we consider here, in particular the graphs are not trees. From [2] it is also known that there exist 2-player, 2-action games for which the expected total payoff of the best correlated equilibrium is higher than the best Nash equilibrium, and we discuss this issue further in Section 7.

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


Learning submodular functions There has been much interest in the machine learning and algorithmic game theory communities on understanding and using submodular functions. Despite this substantial interest, little is known about their learnability from data. Motivated by applications, such as pricing goods in economics, this paper considers PAC-style learning of submodular functions in a distributional setting.A problem instance consists of a distribution on {0, 1} n and a real-valued function on {0, 1} n that is non-negative, monotone, and submodular. We are given poly(n) samples from this distribution, along with the values of the function at those sample points. The task is to approximate the value of the function to within a multiplicative factor at subsequent sample points drawn from the same distribution, with sufficiently high probability. We develop the first theoretical analysis of this problem, proving a number of important and nearly tight results. For instance, if the underlying distribution is a product distribution then we give a learning algorithm that achieves a constant-factor approximation (under some assumptions). However, for general distributions we provide a surprisingÕ(n 1/3 ) lower bound based on a new interesting class of matroids and we also show a O(n 1/2 ) upper bound. Our work combines central issues in optimization (submodular functions and matroids) with central topics in learning (distributional learning and PAC-style analyses) and with central concepts in pseudo-randomness (lossless expander graphs). Our analysis involves a twist on the usual learning theory models and uncovers some interesting structural and extremal properties of submodular functions, which we suspect are likely to be useful in other contexts. In particular, to prove our general lower bound, we use lossless expanders to construct a new family of matroids which can take wildly varying rank values on superpolynomially many sets; no such construction was previously known. This construction shows unexpected extremal properties of submodular functions. What does it mean to "learn a submodular function", and why would one be interested in doing that? To begin, we start by defining what submodular functions are. Let [n] = {1, . . . , n} be a ground set and let f : 2. [n] → R be a set function. This function is called submodular if. Submodularity is in many ways similar to concavity of functions defined on R n . For example, concavity of differentiable functions is equivalent to gradient monotonicity, and submodularity is equivalent to monotonicity of the marginal values:. ( 1.2) for all A ⊆ B ⊆ [n] and i ∈ B. This inequality reflects a natural notion of "diminishing returns", which explains why submodularity has long been a topic of interest in economics [35] . Submodular functions have also been studied for decades in operations research and combinatorial optimization [13] , as they arise naturally in the study of graphs, matroids, covering problems, facility location problems, etc. More recently, submodular functions have become key concepts in both the machine learning and algorithmic game theory communities. For example, submodular functions have been used to model bidders' valuation functions in combinatorial auctions [18, 26, 6] , for solving feature selection problems in graphical models [24] , and for solving various clustering problems [29] . In fact, submodularity has been the topic of several tutorials and workshops at recent major conferences in machine learning [1, 25] .. The Model. So what does it mean to "learn a submodular function"? Our definition comes from learning theory, where the goal of learning is to predict the future based on past observations. One successful approach to formalizing this goal is Valiant's PAC model [37] . However, the PAC model is primarily for learning Booleanvalued functions, such as threshold functions and low-depth circuits [37, 23] . For real-valued functions, it seems appropriate to change the model by ignoring small-magnitude errors in the predicted values. Our results on learning submodular functions are presented in this new model, which we call the PMAC model: this abbreviation stands for "Probably Mostly Approximately Correct".. In this model, a learning algorithm is given a set S of polynomially many labeled examples drawn i.i.d. from some fixed, but unknown, distribution D over points in 2 [n] . The points are labeled by a fixed, but unknown, target function f * : 2. [n] → R+. The goal is to output a hypothesis function f such that, with large probability over the choice of examples, the set of points for which f is a good approximation for f * has large measure with respect to D. More formally,. where f is the output of the learning algorithm when given inputs { (xi, f * (xi)) } i=1,2,... and the approximation ratio α ≥ 1 allows for multiplicative error in the function values. In our model, one must approximate the value of a function on a set of large measure, with high confidence. In contrast, the traditional PAC model requires one to predict the value exactly on a set of large measure, with high confidence. The PAC model is the special case of our model with α = 1.. Motivation. So why would one want to learn a submodular function in this manner? Our work has multiple motivations. From a foundational perspective, submodular functions form a broad class of important functions, so studying their learnability allows us to understand their structure in a new way. To draw a parallel to the Boolean-valued case, a class of comparable breadth is the class of monotone Boolean functions, which have been intensively studied [11, 9, 3] .. From an applications perspective, algorithms for learning submodular functions could be very useful in some of the applications where these functions arise. A classic example is pricing bundles of goods. For example, a software company which produces a software suite typically produces several bundles, each of which is a subset of the software programs in the suite. Further examples abound: automobile manufacturers produce a few trim lines of their vehicles with various added options; supermarkets sell bundles of condiments, variety packs of cereal, etc. From an economic standpoint, the central question here is bundle pricing: how should a company decide which bundles to sell, and how should they choose their prices? This is an active area of research in management science and microeconomic theory [18, 40, 5] .. There has been much work on methods for designing optimally priced bundles, in various models. These methods typically make two assumptions [18] . First, the consumer's valuations for all possible bundles are known. Second, the consumer's valuations exhibit economies of scale (e.g., subadditivity or submodularity). Our work is motivated by the observation that this first assumption is entirely unrealistic, both computationally and pragmatically, since the number of bundles is exponential in the number of goods.. We propose learning of submodular functions as an approach to make this first assumption more realistic. To justify our proposal, note that corporations typically possess large amounts of data acquired from past consumer purchases [31] . This suggests that passive supervised learning is a realistic model for learning consumer valuations. Next, we note that valuations are real numbers, so it may not be possible to learn them exactly. This suggests that allowing solutions with multiplicative error is appropriate for this problem. Thus, the problem of learning submodular functions in the PMAC model is a good fit for the real-world problem of learning consumer valuations. Learning real-valued functions and the PMAC Model. In the machine learning literature [19, 38] , learning real-valued functions (in the distributional learning setting) is often addressed by considering the squared error loss 2 . However, this does not distinguish between the case of having low error on most of the distribution and high error on just a few points, versus moderately high error everywhere. Thus, a lower bound for the squared error loss is not so meaningful. In comparison, the PMAC model allows for more fine-grained control with separate parameters for the amount and extent of errors.. Learning Submodular Functions. To our knowledge, there is no prior work on learning submodular functions in a natural distributional PAC style learning setting. The most relevant work is a paper of Goemans et al. [15] , which considers the problem of "approximating submodular functions everywhere". That paper considers the algorithmic problem of efficiently finding a function which approximates a submodular function at every point of its domain. They give an algorithm which achieves an approximation factor O( √ n), and also showΩ( √ n) inapproximability. Their algorithm adaptively queries the target function at points of its choice, and the hypothesis it produces must approximate the target function at every point. 3 In contrast, our PMAC model falls into the more widely studied passive supervised learning setting [4, 23, 37, 38] , which is more relevant for our motivating application to bundle pricing.. Our algorithm for PMAC-learning under general distributions and the Goemans et al. algorithm both rely on the structural result (due to Goemans et al.) that monotone, submodular functions can be approximated by the square root of a linear function to within a factor √ n. In both cases, the challenge is to find this linear function. The Goemans et al. algorithm is very sophisticated: it gives an intricate combinatorial algorithm to approximately solve a certain convex program which produces the desired function. Additionally, it requires query access to the target and so does not apply to our framework. On the other hand, our algorithm is very simple: given the structural result, we can reduce our problem to that of learning a linear separator, which is easily solved by linear programming. Moreover, our algorithm is noise-tolerant and more amenable to extensions; we elaborate on this in Section 5.. On the other hand, our lower bound is significantly more involved than the lower bound of Goemans et al. [15] . (Their lower bound was slightly improved by Svitkina and Fleischer [34] .) These previous lower bounds also use matroids of the form in Eq. (1.3), although they only need such matroids for the easy case k = 1. Handling the case k = n ω(1) makes our matroid construction much more intricate. Essentially, Goemans et al. only show worst-case inapproximability, whereas we need to show average-case inapproximability. A similar situation occurs with Boolean functions, where lower bounds for distributional learning are typically much harder to show than lower bounds for exact learning. For instance, even conjunctions are hard to learn in the exact learning model, and yet they are trivial to PAC-learn. Proving a lower bound for PAClearning requires exhibiting some fundamental complexity in the class of target functions, especially when one does not restrict the form of the hypothesis function. It is precisely this phenomenon which makes our lower bound challenging to prove.

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