Does the algorithm start with a fully specified deep network?

No. Its defining approach is to grow the structure during learning.

Is a faithful hidden layer already a correct classifier?

Not necessarily. It preserves class distinctions, but later processing may still be needed to produce the desired output.

Can adding units solve contradictory labels?

Not when exactly the same input must deterministically produce two different target values. The contradiction must be addressed in the task specification.

AI Glossary / Graphs and symbolic AI / Advanced

Tiling

Definition

In the neural-network context used here, tiling is a constructive learning method that grows a feedforward network by adding units and layers. It builds internal representations that preserve distinctions between training examples requiring different outputs.

What it means

Instead of choosing a complete network architecture in advance and only adjusting its weights, a constructive method lets training influence the architecture itself. The tiling algorithm of Marc Mezard and Jean-Pierre Nadal is a historical example, developed for learning Boolean mappings with threshold units.

A central idea is faithfulness: if two training inputs need different final answers, an intermediate layer must not represent them identically. Once that distinction is lost, later deterministic layers cannot recover it from that representation alone.

The glossary's source context is network growth. A different use of tiling appears in reinforcement-learning tile coding, where overlapping partitions represent continuous variables. Image tiling and hardware memory tiling are also different topics. Their shared word does not make their algorithms interchangeable.

How it works

The method starts by training a main, or master, unit to classify the available examples. If errors remain, extra ancillary units split groups whose current internal representations mix different target classes. A layer is completed when its representation is faithful to the training task.

The next layer works with those representations. In the original construction, the master unit can be chosen to reduce the training error relative to the preceding master unit. Repeating this process eventually reaches a correct output on a finite, consistently labeled Boolean training set under the method's stated construction.

That is a training-set result. It is not a promise of the smallest possible network, the fastest training process or accurate predictions on unseen inputs. Contradictory labels for identical inputs prevent perfect deterministic fitting unless the problem representation or target is changed. Practical resource limits also matter.

Three practical examples

An AI class studies how a network can acquire additional hidden units when one linear separator cannot represent a Boolean task.
A research comparison measures network size, training error and held-out performance for a constructive learner and a fixed architecture.
A debugging exercise checks whether intermediate representations collapse two differently labeled training examples into the same code.

Think of it this way

Think of sorting documents into trays before a second clerk assigns final categories. If documents that must receive different categories become indistinguishable within the tray description, the second clerk lacks the information needed to separate them.

Adding more descriptive tray labels can preserve useful distinctions. The analogy explains faithful representation, not the mathematical training procedure. A real tiling network uses learned numerical weights and binary activations, not clerks consulting document contents outside the represented inputs.

Worked example

Preserving distinctions for XOR

Consider four input pairs: (0,0), (0,1), (1,0) and (1,1). Their XOR targets are respectively zero, one, one and zero. A single linear threshold unit cannot represent this mapping.

As an original representation illustration, let one hidden unit compute OR and another compute AND. Their two-bit hidden codes are then (0,0), (1,0), (1,0) and (1,1). The two middle inputs share a code, which is acceptable because their targets match. No code is shared by examples with different targets.

An output unit can now return one when h1 minus h2 is at least one. Its scores are zero, one, one and zero, exactly matching XOR. If the hidden layer contained only OR, the last three inputs would all look identical despite different targets.

This hand-designed example checks the representation principle. It is not a claimed trace of the original tiling algorithm or evidence that its training procedure always finds these particular units.

Illustrative case study

Fictional case: Maple Learning Lab compares two small classifiers for a teaching dataset. A fixed single-unit model leaves errors, so the researchers investigate a constructive network that can grow.

Their first report highlights perfect training accuracy. A reviewer asks whether examples with identical features have conflicting labels, whether the reported network includes every added unit, and how performance changes on withheld examples. The team discovers that a newly merged dataset contains inconsistent duplicate labels.

It corrects the data specification, records the network size and keeps an untouched evaluation set. The project becomes a useful demonstration of architectural growth and representation rather than an unsupported claim that zero training error proves business readiness. The case and its workflow are invented.

Common misconceptions and mistakes

Tiling always means a grid over input space. That is a different usage; this entry concerns constructive network growth.
Faithfulness requires a unique code for every training example. Examples with the same target may share a code.
Guaranteed training convergence guarantees generalization. Fitting the presented mapping and predicting unseen cases are separate questions.

Business use

The immediate business value is conceptual: architecture is another modeling choice, and increasing capacity can remove training errors without resolving data quality or generalization problems.

When assessing a growing-network proposal, ask what triggers growth, what stops it, how many units and connections result, and which evaluation data remained separate. Compare total training cost and inference cost under the same workload. Historical convergence arguments should be checked against the actual implementation and its assumptions before they are used in a product claim.

Check your understanding

Two inputs require different labels but have identical outputs at every unit of an intermediate layer. Can a deterministic later layer that sees only those outputs distinguish them?

Answer: No. Its inputs are identical. The intermediate representation must preserve some relevant distinction.

Related terms

Learn next

Read threshold function and hidden layer. Recalculate the OR-and-AND codes in the worked example, then explain why merging the two middle XOR inputs is harmless while merging the last three is not.

Sources

Mezard and Nadal, Learning in feedforward layered networks: the tiling algorithm, original 1989 paper hosted by the author: https://www.marcmezard.fr/wp-content/uploads/2019/01/89_MN_JPA.pdf

Russell and Norvig, author-hosted chapter discussing constructive tiling and its attribution: https://aima.cs.berkeley.edu/newchap20.pdf

Relevant construction, faithfulness and convergence passages were read. The XOR construction is an original explanatory example; no reproduction of the paper's experiments is claimed.

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