# data point

*Entry of the [Dictionary of Applied Machine Learning](https://dictionaryofml.org) (course edition), by Alexander Jung, Aalto University. CC BY 4.0. Updated 2026-10-28. [HTML](https://dictionaryofml.org/terms/datapoint.html) · [typeset PDF](https://dictionaryofml.org/terms/datapoint.pdf) · [LaTeX](https://dictionaryofml.org/terms/datapoint.tex) · [Python demo](https://dictionaryofml.org/terms/datapoint.py) · [notebook](https://dictionaryofml.org/terms/datapoint.ipynb)*

Math is LaTeX with the book's macros expanded, so no notation package is needed to read it.

## Abstract

A data point is the elementary information-carrying unit on which machine learning methods operate. The properties of a data point fall into two categories: features, which are easily measurable or computable, and labels, which are higher-level facts that typically can only be determined using human expertise. Whether a given property is treated as a feature or a label is a design choice that depends on the machine learning application.

## Definition

An image of a cow herd (Fig. 1), a patient in a clinic, and a protein are objects that a machine learning method processes. A data point is an object that conveys information (Cover and Thomas, 2006); such objects are the elementary units on which machine learning methods operate. Examples include students, radio signals, trees, images, RVs, real numbers, or proteins. Data points of the same type are described by two categories of properties.

The first category includes features that are measurable or computable properties of a data point. They can be automatically extracted or computed using sensors, computers, or other data collection systems. The second category includes labels that are higher-level facts, or quantities of interest, that typically require human expertise or domain knowledge to determine rather than being directly measurable. Machine learning aims to predict the label of a data point from its features.

Fig. 1 shows an image as an example of a data point. Several properties of the image can serve as features: the color intensities $x_{1}, \ldots, x_{d}$ of all image pixels, the timestamp $x_{d+1}$ of the image capture, and the spatial location $x_{d+2}$ of the image capture. Higher-level facts can serve as labels: the number of cows $y_{1}$, the number of wolves $y_{2}$, and the condition of the pasture $y_{3}$ (e.g., healthy or overgrazed).

![Figure 1 of the entry datapoint](https://dictionaryofml.org/terms/fig/datapoint-fig1.svg)

*Figure 1: Illustration of a data point consisting of an image. Different properties of the image can serve as features, such as the average color intensities of the image pixels. Higher-level facts about the image, such as the presence of different object categories, can serve as labels*

For a data point that represents a patient, a feature could be the body weight, while the label could be the presence of cancer, the fact of interest. Determining this label requires a medical expert (or even a committee of experts) to examine the patient. A property treated as a label in one setting (e.g., a cancer diagnosis) may serve as a feature in another, where reliable automation (e.g., image analysis) allows the property to be computed without human intervention.

Both kinds of properties are error-prone. For many machine learning applications, it is rarely possible to access the true labels. For example, a medical expert's diagnosis can be wrong, making it a noisy proxy for the patient's true condition. Errors of this kind, called label noise, arise whenever a label is only a proxy for the true quantity of interest, whether that proxy is produced by human judgment or by an automated system.

Features are also a source of error within a machine learning method. A feature is obtained by measuring or computing a property of a data point. Physical and chemical sensing devices have a finite measurement uncertainty (Kimothi, 2002), and computing a feature on a finite computer introduces rounding errors (Golub and Loan, 2013, Sect. 2.7). These imperfections make the recorded value deviate from the true feature value. This discrepancy is feature noise.

Machine learning methods should be robust to both feature and label noise. Their predictions on new data points should not degrade sharply when some features or labels in the training set are corrupted. The EU AI Act reflects this concern: training data must, to the best extent possible, be ``free of errors'' (European Parliament and Council of the European Union, 2024, Art. 10(3)), and high-risk AI systems must attain an appropriate level of robustness (European Parliament and Council of the European Union, 2024, Art. 15(1)).

Robustness to feature and label noise can be improved by data augmentation, which deliberately perturbs the features or the label of a training data point (see Fig. 2). Training on such perturbed data points acts as regularization and reduces the influence of corrupted feature or label values on the learned hypothesis.

![Figure 2 of the entry datapoint](https://dictionaryofml.org/terms/fig/datapoint-fig2.svg)

*Figure 2: Data augmentation surrounds a data point (filled circle) with a cloud of perturbed copies (open circles). It is a form of regularization that makes a machine learning method less sensitive to feature and label noise*

**See also:** [data](https://dictionaryofml.org/terms/data.html), [feature](https://dictionaryofml.org/terms/feature.html), [label](https://dictionaryofml.org/terms/label.html), [dataset](https://dictionaryofml.org/terms/dataset.html), [data augmentation](https://dictionaryofml.org/terms/dataaug.html), robustness, EU AI Act.

## References

- Cover and Thomas (2006). Elements of Information Theory. *Wiley*. <https://doi.org/10.1002/047174882X>
- Kimothi (2002). The Uncertainty of Measurements: Physical and Chemical Metrology Impact and Analysis. *American Society for Quality (ASQ)*.
- Golub and Loan (2013). Matrix Computations. *The Johns Hopkins Univ. Press*.
- European Parliament and Council of the European Union (2024). Regulation (EU) 2024/1689 of the European Parliament and of the Council of 13 June 2024 laying down harmonised rules on artificial intelligence and amending Regulations (EC) No 300/2008, (EU) No 167/2013, (EU) No 168/2013, (EU) 2018/858, (EU) 2018/1139 and (EU) 2019/2144 and Directives 2014/90/EU, (EU) 2016/797 and (EU) 2020/1828 (Artificial Intelligence Act) (Text with EEA relevance). <https://eur-lex.europa.eu/eli/reg/2024/1689/oj/eng>

## Cite this entry

```bibtex
@incollection{dictml_datapoint,
  author = {Jung, Alexander},
  editor = {Olioumtsevits, Konstantina and Schnoor, Ekkehard},
  title = {data point},
  booktitle = {Dictionary of Applied Machine Learning},
  publisher = {Springer Nature Singapore},
  year = {2026},
  doi = {10.1007/978-981-92-2012-0_9-1},
  isbn = {978-981-92-2012-0},
  url = {https://doi.org/10.1007/978-981-92-2012-0_9-1}
}
```