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2011-08-03 08:51:05|  分类: 默认分类 |  标签: |举报 |字号 订阅

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3 Achievements



In this section we will sketch in broad terms the state of the art in building systems for generalization and recognition. In practical applications it is not the primary goal to study the way of bridging the gap between observations and concepts in a scientific perspective. Still, we can learn a lot from the heuristic solutions that are created to assist the human analyst performing a recognition task. There are many systems that directly try to imitate the decision making process of a human expert, such as an operator guarding a chemical process, an inspector supervising the quality of industrial production or a medical doctor deriving a diagnosis from a series of medical tests. On the basis of systematic interviews the decision making can become explicit and imitated by a computer program: an expert system [54]. The possibility to improve such a system by learning from examples is usually very limited and restricted to logical inference that makes the rules as general as possible, and the estimation of the thresholds on observations. The latter is needed as the human expert is not always able to define exactly what he means, e.g. by ‘an unusually high temperature’.



In order to relate knowledge to observations, which are measurements in automatic systems, it is often needed to relate knowledge uncertainty to imprecise, noisy, or generally invalid measurements. Several frameworks have been developed to this end, e.g. fuzzy systems [74], Bayesian belief networks[42] and reasoning under uncertainty [82]. Characteristic for these approaches is that the given knowledge is already structured and needs explicitly defined parameters of uncertainty. New observations may adapt these parameters by relating them to observational frequencies. The knowledge structure is not learned; it has to be given and is hard to modify. An essential problem is that the variability of the external observations may be probabilistic, but the uncertainty in knowledge is based on ‘belief’ or ‘fuzzy’ definitions. Combining them in a single mathematical framework is disputable [39].



In the above approaches either the general knowledge or the concept underlying a class of observations is directly modeled. In structural pattern recognition [26, 65] the starting point is the description of the structure of a single object. This can be done in several ways, e.g. by strings, contour descriptions, time sequences or other order-dependent data. Grammars that are inferred from a collection of strings are the basis of a syntactical approach to pattern recognition [26]. The incorporation of probabilities, e.g. needed for modeling the measurement noise, is not straightforward. Another possibility is the use of graphs. This is in fact already a reduction since objects are decomposed into highlights or landmarks, possibly given by attributes and also their relations, which may be attributed as well. Inferring a language from graphs is already much more difficult than from strings. Consequently, the generalization from a set of objects to a class is usually done by finding typical examples, prototypes, followed by graph matching [5, 78] for classifying new objects.



Generalization in structural pattern recognition is not straightforward. It is often based on the comparison of object descriptions using the entire available training set (the nearest neighbor rule) or a selected subset (the nearest prototype rule). Application knowledge is needed for defining the representation(strings, graphs) as well as for the dissimilarity measure to perform graph matching [51, 7]. The generalization may rely on an analysis of the matrix of dissimilarities, used to determine prototypes. More advanced techniques using the dissimilarity matrix will be described later.



The 1-Nearest-Neighbor Rule (1-NN) is the simplest and most natural classification rule. It should always be used as a reference. It has a good asymptotic performance for metric measures [10, 14], not worse than twice the Bayes error, i.e. the lowest error possible. It works well in practice for finite training sets. Fig. 3 shows how it performs on the Iris data set in comparison to the linear and quadratic classifiers based on the assumption of normal distributions [27]. Thek-NN rule, based on a class majority vote over the nearest neighbors in the training set, is, like the Parzen classifier, even Bayes consistent. These classifiers approximate the Bayes error for increasing training sets [14, 27].



However, such results heavily rely on the assumption that the training examples are identically and independently drawn (iid) from the same distribution as the future objects to be tested. This assumption of a fixed and stationary distribution is very strong, but it yields the best possible classifier. There are, however, other reasons, why it cannot be claimed that pattern recognition is solved by these statistical tools. The 1-NN and k-NN rules have to store the entire training set. The solution is thereby based on a comparison with all possible examples, including ones that are very similar, and asymptotically identical to the new objects to be recognized. By this, a class or

a concept is not learned, as the decision relies on memorizing all possible instances. There is simply no generalization.



Other classification procedures, giving rise to two learning curves shown in Fig. 3, are based on specific model assumptions. The classifiers may perform well when the assumptions hold and may entirely fail, otherwise. An important observation is that models used in statistical learning procedures have almost necessarily a statistical formulation. Human knowledge, however, certainly in daily life, has almost nothing to do with statistics. Perhaps it is hidden in the human learning process, but it is not explicitly available in the context of human recognition. As a result, there is a need to look for effective model assumptions that are not phrased in statistical terms.



In Fig. 3 we can see that a more complex quadratic classifier performs initially worse than the other ones, but it behaves similarly to a simple linear classifier for large training sets. In general, complex problems may be better solved by complex procedures. This is illustrated in Fig. 4, in which the resulting error curves are shown as functions of complexity and training size.



Like in Fig. 3, small training sets require simple classifiers. Larger training sets may be used to train more complex classifiers, but the error will increase, if pushed too far. This is a well-known and frequently studied phenomenon in relation to the dimensionality (complexity) of the problem. Objects described by many features often rely on complex classifiers, which may thereby lead to worse results if the number of training examples is insufficient. This is the curse of dimensionality, also known as the Rao’s paradox or the peaking phenomenon [44, 45]. It is caused by the fact that the classifiers badly generalize,due to a poor estimation of their parameters or their focus/adaptation to the noisy information or irrelevant details in the data. The same phenomenon can be observed while training complex neural networks without taking proper precautions. As a result, they will adapt to accidental data configurations, hence they willovertrain. This phenomenon is also well known outside the pattern recognition field. For instance, it is one of the reasons one has to be careful with extensive mass screening in health care: the more diseases and their relations are considered (the more complex the task), the more people will we be unnecessarily sent to hospitals for further examinations.




An important conclusion from this research is that the cardinality of the set of examples from which we want to infer a pattern concept bounds the complexity of the procedure used for generalization. Such a method should be simple if there are just a few examples. A somewhat complicated concept can only be learnt if sufficient prior knowledge is available and incorporated in such a way that the simple procedure is able to benefit from it.



An extreme consequence of the lack of prior knowledge is given by Watanabe as the Ugly Duckling Theorem [75]. Assume that objects are described by a set of atomic properties and we consider predicates consisting of all possible logic combinations of these properties in order to train a pattern recognition system. Then, all pairs of objects are equally similar in terms of the number of predicates they share. This is caused by the fact that all atomic properties, their presence as well as their absence, have initially equal weights.As a result, the training set is of no use. Summarized briefly, if we do not know anything about the problem we cannot learn (generalize and/or infer) from  observations.

一个缺乏先验知识的极端结论是Watanabe的丑小鸭定理(Ugly Duckling Theorem)[75].假设对象被描述成一个原子性质集,对这些性质进行所有可能的逻辑合并,合并后再进行组合成对象的属性,以此来训练一个模式识别系统。于是任何一对对对象在一些共有的属性上是同等相似的。这是由于对于所有原子性质,跟对象的存在与否无关,初始时二者都具有一样的权值。由此,这里训练集是没有用的。简要地总结一下,就是如果我们对这个问题什么都不了解,我们不可能从观察中学会(推广或推导)。


An entirely different reasoning pointing to the same phenomenon is the No-Free-Lunch Theoremformulated by Wolpert [81]. It states that all classifiers perform equally well if averaged over all possible classification problems.This also includes a random assignment of objects to classes. In order to understand this theorem it should be realized that the considered set of all possible classification problems includes all possible ways a given data set can be distributed over a set of classes. This again emphasizes that learning cannot be successful without any preference or knowledge.

对这相同现象有一个完全不同的论证:Wolpert的没有免费的午餐定理(No-Free-Lunch Therorem)[81]。他指出如果平衡所有可能的分类问题,则所有的分类器的性能是一样的,这也包括指对一个随机的分类方法。要理解这个定理则必须明白:对于所有可能的分类问题,包括所有的可能分类方法,总有一组数据可以被用于对一组类别的识别。这又强调了没有进行优化选择或缺少先验知识,这样的学习是不会成功的。


In essence, it has been established that without prior or background knowledge,no learning, no generalization from examples is possible. Concerning specific applications based on strong models for the classes, it has been shown that additional observations may lower the specified gaps or solve uncertainties in these models. In addition, if these uncertainties are formulated in statistical terms, it will be well possible to diminish their influence by a statistical analysis of the training set. It is, however, unclear what the minimum prior knowledge is that is necessary to make the learning from examples possible.This is of interest if we want to uncover the roots of concept formation, such as learning of a class from examples. There exists one principle, formulated at the very beginning of the study of automatic pattern recognition, which may point to a promising direction. This is the principle of compactness [1], also phrased as a compactness hypothesis. It states that we can only learn from examples or phenomena if their representation is such that small variations in these examples cause small deviations in the representation. This demands that the representation is based on a continuous transformation of the real world objects or phenomena. Consequently, it is assumed that a sufficiently small variation in the original object will not cause the change of its class membership. It will still be a realization of the same concept. Consequently,we may learn the class of objects that belong to the same concept by studying the domain of their corresponding representations.



The Ugly Duckling Theorem deals with discrete logical representations.These cannot be solved by the compactness hypothesis unless some metric is assumed that replaces the similarity measured by counting differences in predicates.The No-Free-Lunch Theorem clearly violates the compactness assumption as it makes object representations with contradictory labelings equally probable. In practice, however, we encounter only specific types of problems.



Building proper representations has become an important issue in pattern recognition [20]. For a long time this idea has been restricted to the reduction of overly large feature sets to the sizes for which generalization procedures can produce significant results, given the cardinality of the training set. Several procedures have been studied based on feature selection as well as linear and nonlinear feature extraction [45]. A pessimistic result was found that about any hierarchical ordering of (sub)space separability that fulfills the necessary monotonicity constraints can be constructed by an example based on normal distributions only [11]. Very advanced procedures are needed to find such ‘hidden’ subspaces in which classes are well separable [61]. It has to be doubted, however, whether such problems arise in practice, and whether such feature selection procedures are really necessary in problems with finite sample sizes. This doubt is further supported by an insight that feature reduction procedures should rely on global and not very detailed criteria if their purpose is to reduce the high dimensionality to a size which is in agreement with the given training set.



Feed-forward neural networks are a very general tool that, among others, offer the possibility to train a single system built between sensor and classification [4, 41, 62]. They thereby cover the representation step in the input layer(s) and the generalization step in the output layer(s). These layers are simultaneously optimized. The number of neurons in the network should be sufficiently large to make the interesting optima tractable. This, however,brings the danger of overtraining. There exist several ways to prevent that by incorporating some regularization steps in the optimization process. This replaces the adaptation step in Fig. 1. A difficult point here, however, is that it is not sufficiently clear how to choose regularization of an appropriate strength.The other important application of neural networks is that the use of various regularization techniques enables one to control the nonlinearity of the resulting classifier. This gives also a possibility to use not only complex, but also moderately nonlinear functions. Neural networks are thereby one of the most general tools for building pattern recognition systems.



In statistical learning, Vapnik has rigorously studied the problem of adapting the complexity of the generalization procedure to a finite training set[72, 73]. The resulting Vapnik-Chervonenkis(VC) dimension, a complexity measure for a family of classification functions, gives a good insight into the mechanisms that determine the final performance (which depends on the training error and the VC dimension). The resulting error bounds, however,are too general for a direct use. One of the reasons is that, like in the No-Free-Lunch Theorem, the set of classification problems (positions and labeling of the data examples) is not restricted to the ones that obey the compactness assumption.



One of the insights gained by studying the complexity measures of polynomial functions is that they have to be as simple as possible in terms of the number of their free parameters to be optimized. This was already realized by Cover in 1965 [9]. Vapnik extended this finding around 1994 to arbitrary non-linear classifiers [73]. In that case, however, the number of free parameters is not necessarily indicative for the complexity of a given family of functions,but the VC dimension is. In Vapnik’s terms, the VC dimension reflects the flexibility of a family of functions (such as polynomials or radial basis functions)to separate arbitrarily labeled and positioned n-element data in a vector space of a fixed dimension. This VC dimension should be finite and small to guarantee the good performance of the generalization function.



This idea was elegantly incorporated to the Support Vector Machine (SVM) [73], in which the number of parameters is as small as a suitably determined subset of the training objects (the support vectors) and into independent of the dimensionality of the vector space. One way to phrase this principle is that the structure of the classifier itself is simplified as far as possible (following the Occam’s razor principle). So, after a detor along huge neural networks possibly having many more parameters than training examples, pattern recognition was back to the small-is-beautiful principle, but now better understood and elegantly formulated.



The use of kernels largely enriched the applicability of the SVM to nonlinear decision functions [66, 67, 73]. The kernel approach virtually generates nonlinear transformations of the combinations of the existing features. By using the representer theorem, a linear classifier in this nonlinear feature space can be constructed, because the kernel encodes generalized inner products of the original vectors only. Consequently, well-performing nonlinear classifiers built on training sets of almost any size in almost any feature space can be computed by using the SVM in combination with the ‘kernel trick’ [66].



This method has still a few limitations, however. It was originally designed for separable classes, hence it suffers when high overlap occurs. The use of slack variables, necessary for handling such an overlap, leads to a large number of support vectors and, consequently, to a large VC dimension. In such cases,other learning procedures have to be preferred. Another difficulty is that the class of admissible kernels is very narrow to guarantee the optimal solution.A kernel has to be (conditionally) positive semidefinite (cpd) functions of two variables as only then it can be interpreted as a generalized inner product in reproducing kernel Hilbert space induced by K. Kernels were first considered as functions in Euclidean vector spaces, but they are now also designed to handle more general representations. Special-purpose kernels are defined in a number of applications such as text processing and shape recognition, in which good features are difficult to obtain. They use background knowledge from the application in which similarities between objects are defined in such a way that a proper kernel can be constructed. The difficulty is, again, the strong requirement of kernels as being cpd.



The next step is the so-called dissimilarity representation [56] in which general proximity measures between objects can be used for their representation.The measure itself may be arbitrary, provided that it is meaningful for the problem. Proximity plays a key role in the quest for an integrated structural and statistical learning model, since it is a natural bridge between these two approaches [6, 56]. Proximity is the basic quality to capture the characteristics of a set objects forming a group. It can be defined in various ways and contexts, based on sensory measurements, numerical descriptions, sequences, graphs, relations and other non-vectorial representations, as well as their combinations. A representation based on proximities is, therefore,universal.



Although some foundations are laid down [56], the ways for effective learning from general proximity representations are still to be developed. Since measures may not belong to the class of permissable kernels, the traditional SVM, as such, cannot be used. There exist alternative interpretations of indefinite kernels and their relation to pseudo-Euclidean and Krein spaces[38, 50, 55, 56, 58], in which learning is possible for non-Euclidean representations.In general, proximity representations are embedded into suitable vector spaces equipped with a generalized inner product or norm, in which numerical techniques can either be developed or adapted from the existing ones. It has been experimentally shown that many classification techniques may perform well for general dissimilarity representations.


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