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Topic: Linearly separable

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	[No title]
		Neural networks simply take advantage of the fact that what is not linearly separable in k dimensions is linearly separable in higher dimensions, often in k+1 dimensions.
		Without using any new information, if the original data is not linearly separable, multi-layer neural networks create artificial inputs which are polynomials of the original inputs.
		Obviously, there is no way to separate the two classes with a single line, and therefore a one-layer perceptron will fail to learn the two classes of data.
	www.cs.mcgill.ca /~psavad/webproject/whatlearn.html (742 words)

	Neural Networks - Architecture
		Now let us extend the discussion into the separation of points into two regions that are not linearly separable.
		With four perceptrons that are independent of each other in the hidden layer, the point is classified into 4 pairs of linearly separable regions, each of which has a unique line separating the region.
		The top perceptron performs logical operations on the outputs of the hidden layers so that the whole network classifies input points in 2 regions that might not be linearly separable.
	cse.stanford.edu /classes/sophomore-college/projects-00/neural-networks/Architecture/feedforward.html (688 words)

	Pattern Classification Via Linear Programming
		Linearly Separable: 2 points sets H and M are linearly separable if and only if there exists a hyperplane that separates H and M. Linearly Inseparable: When 2 points sets H and M are not linearly separable.
		H and M are linearly separable since all of H's points are on one side of the hyperplane
		H and M are linearly separable if and only if the optimal value of (LP) is zero.
	cgm.cs.mcgill.ca /~beezer/cs644/main.html (1681 words)

	Neural network - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-09-19)
		It calculates the errors between calculated output and sample output data, and uses this to create an adjustment to the weights, thus implementing a form of gradient descent.
		Single-unit perceptrons are only capable of learning linearly separable patterns; in 1969 in a famous monograph entitled Perceptrons by Marvin Minsky and Seymour Papert showed that it was impossible for a single-layer perceptron network to learn an XOR function.
		While a feed-forward network propagates data linearly from input to output, RNs also propagate data from later processing stages to earlier stages.
	encyclopedia.worldsearch.com /neural_network.htm (2810 words)

	Perceptrons Revisited (Minsky & Papert)
		Functions that have an order of 1 are called ``linearly separable'' and can be solved by a single layer perceptron.
		These types of problems are termed ``linearly inseparable'' and require a layer of processing units between the input and output units.
		At the time, however, there was no reliable method of training this intermediate level, and therefore perceptrons were limited to being trained on linearly separable problems only.
	neuron-ai.tuke.sk /NCS/VOL1/P3_html/node20.html (768 words)

	Linearly Disjoint
		If l/k and m/k are field extensions, and l is a finite separable extension, and m is disjoint from the normal closure of l, then l and m are linearly disjoint.
		is finite, and its normal closure is disjoint from m, hence the extensions are linearly disjoint.
		In summary, l and m are linearly disjoint if m is disjoint from the normal closure of l, and either extension is finite.
	www.mathreference.com /fld-sep,lindis.html (895 words)

	Learning in TLUs
		The simplest case of a linearly separable decision problem is one consisting of two sets of points (patterns) in a 2-d vector space that belong to different classes, where the two classes can be separated by a straight line.
		In higher dimensional cases two classes in an N dimensional space are termed "linearly separable" if they are separable by a hyperplane of dimension N-1.
		Clearly not all decision problems are linearly separable: they cannot be solved using a linear decision boundary.
	www.cs.bham.ac.uk /~jlw/sem2a2/Web/LearningTLU.htm (1197 words)

	A Historical Perspective on Neural Networks from Chapter 1: Overview of Artificial Intelligence
		The perceptron was unable to lean to recognize input that was not “linearly separable.” This would prove to be huge obstacle that the neural network would take some time to overcome.
		This table would be considered to be linearly separable.
		The Perception’s inability to solve non-linearly separable problems would prove to be a major obstacle to not only the Perceptron, but the entire field of neural networks.
	www.heatonresearch.com /articles/1/page4.html (1640 words)

	Support Vector classifiers for Land Cover Classification
		These training patterns are said to be linearly separable if a vector w (which determining the orientation of a discriminating plane) and a scalar b (determine offset of the discriminating plane from origin) can be defined so that inequalities (1) and (2) are satisfied.
		The first term in (8) is same as in as in the linearly separable case, and controls the learning capacity, while the second term controls the number of misclassified points.
		Where it is not possible to have a hyperplane defined by linear equations on the training data, the techniques described above for linearly separable data can be extended to allow for non-linear decision surfaces.
	www.gisdevelopment.net /technology/rs/mi03023a.htm (911 words)

	Neural Networks - Neuron
		In the above graphs, the two axes are the inputs which can take the value of either 0 or 1, and the numbers on the graph are the expected output for a particular input.
		However, not all logic operators are linearly separable.
		For instance, the XOR operator is not linearly separable and cannot be achieved by a single perceptron.
	cse.stanford.edu /classes/sophomore-college/projects-00/neural-networks/Neuron (462 words)

	Assignment 7 information
		These data may not be linearly separable (although the documentation for the data set says that they are).
		I will be providing at least one synthetic data set which will be linearly separable.
		you can produce your own synthetic data which is guaranteed to be linearly separable.
	www.cs.rpi.edu /courses/fall01/ai/assign/assign7 (545 words)

	Using TLUs and perceptrons as classifiers (Site not responding. Last check: 2007-09-19)
		Examples from these classes may have been obtained, for example, by capturing images in a framestore; there may be two classes of faces, or we want to separate handwritten characters into numerals and letters.
		That is the two classes (A,B) (C,D) are linearly separable, as too are the classes (A,D) and (B,C).
		Notice that the grouping (A,C) (D,B) would not have worked, since these are not linearly separable, and other arrangements of the four classes in pattern space will require a different set of groupings.
	www.shef.ac.uk /psychology/gurney/notes/l3/section3_2.html (351 words)

	[No title] (Site not responding. Last check: 2007-09-19)
		You may find it helpful to open a separate browser window for the instructions, so you can view them at the same time as the applet window.
		Different cluster dispersions: Place 20 red points (1) in a very narrow cluster (strongly correlated points) and 5 blue points (0) in a very wide cluster in such a way that the classes are linearly separable.
		Imperfectly separable case: Place 10 red points to (1) and 10 blue points (0) in two similar, linearly separable clusters.
	diwww.epfl.ch /mantra/tutorial/english/apb/html (371 words)

	andra
		The question is how many of the 256 Boolean functions with three variables that are linearly separable (can be handled by a perceptron net without hidden units).
		For 3 inputs I think there are 104 linearly separable functions.
		Then 12 ways to pick two corners that are linearly separable (the edges).
	hem.passagen.se /hhic/Ass2.html (945 words)

	rapid detection of color differences in search for a linearly separable target (Site not responding. Last check: 2007-09-19)
		We examined search for a linearly separable target, but we presented different sets of distractors that would require different linear operators in order to isolate the target.
		In addition, performance on the mixed trials (linearly separable) was much better than performance on trials in which all three distractor colors were presented on each trial (not linearly separable).
		The fact that intermixing of distractor sets caused no cost to performance, and the fact that performance in the intermixed condition was better than performance when all three distractor colors were presented, suggest that observers were able to select the appropriate operator for a given distractor set in a bottom-up fashion.
	info.wlu.ca /~wwwpsych/eolds/arvo97.html (302 words)

	► » Can the Adaline Linearly Separate? (Site not responding. Last check: 2007-09-19)
		d(x) in {-1, +1} for each x in the training set E. E was linearly separable.
		Is there a linearly separable input training set the Adaline can't
		that separation is impossible so, like the energizer bunny...
	www.comp-one.net /Can-the-Adaline-Linearly-Separate-6645614.html (389 words)

	IBM Research - Trainable Information Extraction Systems
		In practical NLP applications, data may not always be linearly separable.
		The resulting method converges both in the linearly separable case and in the linearly non-separable case.
		Its numerical stability implies that the new method is more suitable for practical NLP problems that may not be linearly separable.
	www.research.ibm.com /IE (1389 words)

	[No title]
		This dataset is not linearly separable, we’ll never be able to find a line (also called a linear decision surface, or a linear classifier) such that all the + are on one side and all the - are on the other.
		Note that we never know which features will make our data linearly separable, x2 and x2+y2 were lucky guesses, but if we pick many non-linear features (like xy, sin(x), log(x2y), etc), then there is a good chance our data will become linearly separable, even though there is no certainty.
		This is a very powerful set of features; remember in the examples before simply choosing x2 as a feature made the data linearly separable, this particular expansion is more powerful, so we expect even more datasets that were not linearly separable in 2 dimensions to be linearly separable in this feature space.
	www.stanford.edu /class/cs276a/handouts/svmreview.doc (1863 words)

	[No title]
		By analyzing this data you may notice that when input 2 is greater than 0.5 that the data point is of class one and otherwise it is of class two.
		This means that this data is linearly seperable.
		If the two inputs were plotted on an x,y graph the two classes would be separated by the line y=0.5x.
	hercules.ece.utexas.edu /~adrian/rbfvis/homework (1149 words)

	Suggestions for exam questions (Site not responding. Last check: 2007-09-19)
		For each of the following problems explain whether the function is linearly separable.
		You may want to use three dimensional pictures of cubes to visualize whether the functions are linearly separable.
		If a function is separable, give a set of weights and thresholds that solve the problem.
	www.cs.swarthmore.edu /~meeden/nnet/section3_3.html (386 words)

	UCL/CORE - Illés.sem (Site not responding. Last check: 2007-09-19)
		Sufficient optimality criteria for linearly constrained, concave minimization problems is the topic of the talk.
		Phillips and Rosen (1993) in their paper derived sufficient optimality criteria for a slightly different, linearly constrained, concave minimization problem using exponentially many linear programming problems.
		We introduced special test points and using these, for several cases, we are able to show the optimality of the current basic solution.
	www.core.ucl.ac.be:16080 /seminars/Illes.html (162 words)

	[No title] (Site not responding. Last check: 2007-09-19)
		Unfortunately this is only useful if the problem is linearly separable.
		A linearly separable problem is one in which the classes can be separated by a single hyperplane.
		It is often the case that a problem is not linearly separable.To solve these we use a Multi-Layer Perceptron (MLP) where one layer feeds into the next.
	cs-alb-pc3.massey.ac.nz /notes/59318/l7.html (569 words)

	Generalisation in MLPs (Site not responding. Last check: 2007-09-19)
		Easy problem: Set two simple clusters, a red one (1's) and a blue one (0's), of training points linearly separable and well distinct.
		To be realistic, the cross-validation points should be of the same color as training points in the same cluster.
		Non-linearly separable problem: Set 3 blue training points on the left hand side of the space, 6 red training points in the middle and 3 blue training points on the right hand side.
	diwww.epfl.ch /mantra/tutorial/english/general/html (419 words)

	Target Adaptation to Improve the Performance of Least-Squared Classifiers (Site not responding. Last check: 2007-09-19)
		In classifier design, the squared error criterion is often used as an approximation to more relevant cost functions based on the number of classification errors, due to the relative computational ease of least-squares methods.
		This practice results in decision boundaries, which are sub-optimal in terms of classifier accuracy, often failing to separate even linearly separable classes.
		We demonstrate its use with linear least-squares classifiers, and give a bound on the number of iterations required for the special case of linearly separable classes.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/ijcnn/2000/0619/01/0619toc.xml&DOI=10.1109/IJCNN.2000.857821 (223 words)

	An Intuition: Pattern Recognition Via LP
		Problem 2: If H and M are linearly separable, find a separating hyperplane for H and M. Claim: The following linear program solves problem 1 and 2.
		Problem 3: If H and M are not linearly separable, find "the best" hyperplane that separates most of the patterns in H and M. Mangasarian's LP formulation attempts to solve all three problems:
		(3) Else (if the two pattern sets are not linearly separable) compute a "good" hyperplane that splits the two sets as "best" as possible.
	cgm.cs.mcgill.ca /~beezer/cs644/seperable.html (486 words)

	Island Model Genetic Algorithms and Linearly Separable Problems - Whitley, Rana, Heckendorn (ResearchIndex) (Site not responding. Last check: 2007-09-19)
		In the case of the Island Model Genetic Algorithm, it has been informally argued that having multiple subpopulations helps to preserve genetic diversity, since each island can potentially follow a different search trajectory through the search space.
		On the other hand, linearly separable functions have often been used to test Island Model...
		Whitley, S. Rana and R. Heckendorn, "Island Model Genetic Algorithms and Linearly Separable Problems",Proceedings of the AISB Workshop on Evolutionary Computation, 1997 http://citeseer.ist.psu.edu/whitley97island.html More
	citeseer.ist.psu.edu /whitley97island.html (556 words)

	An Approximation Algorithm to Find the Largest Linearly Separable Subset of Training Examples (ResearchIndex) (Site not responding. Last check: 2007-09-19)
		Abstract: We present an approximation algorithm for the NP-hard problem of finding the largest linearly separable subset of examples among a training set.
		We present numerical evidence of its superiority over the Pocket algorithm used by many neural net constructive algorithms.
		1 Introduction In this paper, we present an approximation algorithm for finding the largest linearly separable subset of examples among a training set.
	citeseer.ist.psu.edu /292865.html (228 words)

	Fuzzy patterns recognition (Site not responding. Last check: 2007-09-19)
		Based on the features of level sets, an idea of linearly separable fuzzy clusters is introduced.
		An algorithm is proposed for deriving a decision-making function, based on the technique originally used for the crisp case.
		All algorithms are computer-oriented and can be implemented for the automatic recognition of fuzzy patterns.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/isuma/1995/7126/00/7126toc.xml&DOI=10.1109/ISUMA.1995.527796 (216 words)

	Compact, separable, linearly ordered spaces by Mary Ellen Rudin (Site not responding. Last check: 2007-09-19)
		Compact, separable, linearly ordered spaces by Mary Ellen Rudin
		A proof that a compact, separable, zero-dimensional, monotonically normal space is always the continuous image of a compact linearly ordered space is given.
		The author has granted their consent to include this document in Topology Atlas.
	at.yorku.ca /e/a/p/p/11.htm (54 words)

	JARS.COM - The #1 Java Review Service - Linearly Separable
		Learn how Code Signing certificates are used to secure code and content & how they operate with different software platforms.
		Given any binary switching function of 16 or fewer variables, Linearly will report the minimal verticies to remove to make the problem linearly separable.
		Researchers have been working on this problem for decades without definitive results.
	www.jars.com /science/computers/resource.php/8451 (252 words)

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