What is the difference between a multiclass problem and a multilabel problem?
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I suspect the difference is that in multi-class problems the classes are mutually exclusive, whereas for multi-label problems each label represents a different classification task, but the tasks are somehow related (so there is a benefit in tackling them together rather than separately). For example, in the famous leptograspus crabs dataset there are examples of males and females of two colour forms of crab. You could approach this as a multi-class problem with four classes (male-blue, female-blue, male-orange, female-orange) or as a multi-label problem, where one label would be male/female and the other blue/orange. Essentially in multi-label problems a pattern can belong to more than one class.
A multi-class problem has the assignment of instances to one of a finite, mutually-exclusive collection of classes. As in the example already given of crabs (from @Dikran): male-blue, female-blue, male-orange, female-orange. Each of these is exclusive of the others and taken together they are comprehensive.
One form of a multi-label problem is to divide these into two labels, sex and color; where sex can be male or female, and color can be blue or orange. But note that this is a special case of the multi-label problem as every instance will get every label (that is every crab has both a sex and a color).
Multi-label problems also include other cases that allow for a variable number of labels to be assigned to each instance. For instance, an article in a newspaper or wire service may be assigned to the categories NEWS, POLITICS, SPORTS, MEDICINE, etc. One story about an important sporting event would get an assignment of the label SPORTS; while another, involving political tensions that are revealed by a particular sporting event, might get both the labels SPORTS and POLITICS. Where I am, in the US, the results of the Superbowl are labeled both SPORTS and NEWS given the societal impact of the event.
Note that this form of labeling, with variable numbers of labels, can be recast into a form similar to the example with the crabs; except that every label is treated as LABEL-X or not-LABEL-X. But not all methods require this recasting.
Multiclass classification means a classification task with more than two classes; e.g., classify a set of images of fruits which may be oranges, apples, or pears. Multiclass classification makes the assumption that each sample is assigned to one and only one label: a fruit can be either an apple or a pear but not both at the same time.
Multilabel classification assigns to each sample a set of target labels. This can be thought as predicting properties of a data-point that are not mutually exclusive, such as topics that are relevant for a document. A text might be about any of religion, politics, finance or education at the same time or none of these.
To complement the other answers, here are some figures. One row = the expected output for one sample.
One column = one class (one-hot encoding)
One column = one class
You see that:
As a side note, nothing prevents you from having a multioutput-multiclass classification problem, e.g.: