---
title: "Support-Vector Networks — paper by Cortes and Vapnik (1995)"
description: "It presents the support-vector network, a learning machine that sends inputs through a fixed… 15 references, cited by 578 papers here."
authors: ["Corinna Cortes", "Vladimir Vapnik"]
year: 1995
venue: "Machine Learning"
source: "https://doi.org/10.1007/BF00994018"
doi: "10.1007/bf00994018"
licence: "free access at the publisher/repository (Machine Learning, publishedVersion), no licence stated — read only"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/cortes-1995-support-vector-networks/"
---
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# Support-Vector Networks

**Authors:** Corinna Cortes and Vladimir Vapnik · **Year:** 1995 · **Venue:** [Machine Learning](https://md.research.elamur.ai/free/en/venues/machine-learning/) ([Machine Learning 1995](https://md.research.elamur.ai/free/en/venues/machine-learning/1995/)) · **Publisher:** [Springer Science+Business Media](https://md.research.elamur.ai/free/en/publishers/springer-science-business-media/) · **Licence:** free access at the publisher/repository (Machine Learning, publishedVersion), no licence stated — read only

**Links:** [Publisher's page](https://doi.org/10.1007/bf00994018) · [PDF](https://link.springer.com/content/pdf/10.1007/BF00994018.pdf) · [DOI 10.1007/bf00994018](https://doi.org/10.1007/bf00994018) · [OpenAlex](https://openalex.org/W4239510810)

**Citations:** 58,873 citations (OpenAlex, 2026-10-14) · [578](https://md.research.elamur.ai/free/en/papers/cortes-1995-support-vector-networks/cited-by/) in our library, of the 149,439 papers whose bibliographies we read

## Summary
The paper tackles two-class pattern recognition. It presents the support-vector network, a learning machine that sends inputs through a fixed non-linear mapping into a very high-dimensional feature space and fits a linear decision surface there. Earlier work on optimal (maximum-margin) hyperplanes required training data that could be separated with no errors. This paper removes that restriction with a "soft margin" formulation that tolerates training mistakes through slack variables and still reduces to a quadratic programming problem. The resulting classifier depends only on a small subset of training points, the support vectors. Following Boser, Guyon & Vapnik (1992), the comparison between an input and the support vectors is made directly in input space through a kernel-like function, so the huge feature space never has to be built explicitly. The authors test polynomial decision surfaces up to degree 7 on 256-dimensional inputs in an optical character recognition benchmark and compare them with linear classifiers, k-nearest neighbours and neural networks. The work matters because it gives a principled, convex alternative to neural networks with strong generalization.

## Key points
- The expected test error of a separating optimal hyperplane is bounded by the expected number of support vectors divided by the training set size. The bound does not explicitly depend on feature-space dimension.
- On real data this ratio can fall to about 0.03, and the optimal hyperplane still generalizes in a feature space of roughly a billion dimensions.
- The soft-margin extension penalizes slack with F(u)=u^k. The reported experiments use k=2. With k=1, the dual becomes a quadratic program with box constraints 0 ≤ α ≤ C.
- The solution weight vector is a linear combination of support vectors only, so the classifier can be viewed as a two-layer network.
- The experiments use polynomial kernels up to degree 7 on 256-dimensional OCR inputs. The paper also references NIST Special Database 3.

## Interesting ideas
- **Kernel trick (swap the order of operations):** compare vectors in input space first, then apply the non-linearity. This lets you get very rich feature spaces without ever computing them, and it transfers to any similarity-based model.
- **Support-vector count as a generalization proxy:** the fraction of training points that end up as support vectors gives a cheap, dimension-free estimate of test error. It could be useful for model selection when held-out data is scarce.
- **Soft margin via slack variables:** allowing controlled constraint violation with a penalty weight C turns a brittle "must separate" objective into a robust convex problem. The same pattern works for noisy labels in other tasks.
- **Sparse solution representation:** because the model is stored as a few support vectors with multipliers, its cost depends on how many points matter, not on the feature dimension. This is attractive for compact deployment.
- **Choice of slack penalty:** the authors treat the error penalty F(u) as a design knob, covering linear, squared and least-squares variants. Varying the loss shape like this is a useful lens when tuning any margin-based classifier.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

## Institutions

- [AT&T](https://md.research.elamur.ai/free/en/companies/at-t/) — Company, Bedminster, United States · 40 works · 75,466 citations

📚 Shelves (2): [🖥️ Computer Science](https://md.research.elamur.ai/free/en/shelves/computer-science/) · 16,667 works · [🤖 Artificial Intelligence](https://md.research.elamur.ai/free/en/shelves/computer-science/artificial-intelligence/) · 16,194 works

💡 Concepts (1): [💡 Machine learning](https://md.research.elamur.ai/free/en/concepts/machine-learning/) · 51 works

## Cite

Cite as: Corinna Cortes and Vladimir Vapnik. 1995. Support-Vector Networks. Machine Learning.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@inproceedings{cortes1995support,
  title = {{Support-Vector Networks}},
  author = {Corinna Cortes and Vladimir Vapnik},
  booktitle = {Machine Learning},
  year = {1995},
  doi = {10.1007/bf00994018},
  url = {https://doi.org/10.1007/bf00994018}
}
```

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## References

15 references, 0 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

1. Aizerman, M., Braverman, E., & Rozonoer, L. (1964). Theoretical foundations of the potential function method in pattern recognition learning. Automation and Remote Control, 25:821-837. [find](https://scholar.google.com/scholar?q=Aizerman%2C+M%2E%2C+Braverman%2C+E%2E%2C+%26+Rozonoer%2C+L%2E+%281964%29%2E+Theoretical+foundations+of+the+potential+function+method+in+pattern+recognition+learning%2E+Automation+and+Remote+Control%2C+25%3A821%2D837%2E)
2. Anderson, T.W., & Bahadur, R.R. (1966). Classification into two multivariate normal distributions with different covariance matrices. Ann. Math. Stat., 33:420--431. [find](https://scholar.google.com/scholar?q=Anderson%2C+T%2EW%2E%2C+%26+Bahadur%2C+R%2ER%2E+%281966%29%2E+Classification+into+two+multivariate+normal+distributions+with+different+covariance+matrices%2E+Ann%2E+Math%2E+Stat%2E%2C+33%3A420%2D%2D431%2E)
3. Boser, B.E., Guyon, I., & Vapnik, V.N. (1992). A training algorithm for optimal margin classifiers. In Proceedings of the F(fth Annual Worlaghop of Computational Learning Theory, 5, 144-152, Pittsburgh, ACM. [find](https://scholar.google.com/scholar?q=Boser%2C+B%2EE%2E%2C+Guyon%2C+I%2E%2C+%26+Vapnik%2C+V%2EN%2E+%281992%29%2E+A+training+algorithm+for+optimal+margin+classifiers%2E+In+Proceedings+of+the+F%28fth+Annual+Worlaghop+of+Computational+Learning+Theory%2C+5%2C+144%2D152%2C+Pittsburg)
4. Bottou, L., Cortes, C., Denker, J.S., Drucker, H., Guyon, I., Jackel, L.D., LeCun, Y., Sackinger, E., Simard, P., Vapnik, V., & Miller, U.A. (1994). Comparison of classifier methods: A case study in handwritten digit recognition. Proceedings of 12th International Conference on Pattern Recognition an… [find](https://scholar.google.com/scholar?q=Bottou%2C+L%2E%2C+Cortes%2C+C%2E%2C+Denker%2C+J%2ES%2E%2C+Drucker%2C+H%2E%2C+Guyon%2C+I%2E%2C+Jackel%2C+L%2ED%2E%2C+LeCun%2C+Y%2E%2C+Sackinger%2C+E%2E%2C+Simard%2C+P%2E%2C+Vapnik%2C+V%2E%2C+%26+Miller%2C+U%2EA%2E+%281994%29%2E+Comparison+of+classifier+methods%3A+A+case+study+in+h)
5. Bromley, J., & Sackinger, E. (1991). Neural-network and k-nearest-neighbor classifiers. Technical Report 11359-910819-16TM, AT&T. [find](https://scholar.google.com/scholar?q=Bromley%2C+J%2E%2C+%26+Sackinger%2C+E%2E+%281991%29%2E+Neural%2Dnetwork+and+k%2Dnearest%2Dneighbor+classifiers%2E+Technical+Report+11359%2D910819%2D16TM%2C+AT%26T%2E)
6. Courant, R., & Hilbert, D. (1953). Methods ofMathematical Physics, Interscience, New York. [find](https://scholar.google.com/scholar?q=Courant%2C+R%2E%2C+%26+Hilbert%2C+D%2E+%281953%29%2E+Methods+ofMathematical+Physics%2C+Interscience%2C+New+York%2E)
7. Fisher, R.A. (1936). The use of multiple measurements in taxonomic problems. Ann. Eugenics, 7:111-132. [find](https://scholar.google.com/scholar?q=Fisher%2C+R%2EA%2E+%281936%29%2E+The+use+of+multiple+measurements+in+taxonomic+problems%2E+Ann%2E+Eugenics%2C+7%3A111%2D132%2E)
8. LeCun, Y. (1985). Une procedure d'apprentissage pour reseau a seuil assymetrique. Cognitiva 85: A la Frontiere de l'Intelligence Artificielle des Sciences de la Connaissance des Neurosciences, 599-604, Paris. [find](https://scholar.google.com/scholar?q=LeCun%2C+Y%2E+%281985%29%2E+Une+procedure+d%27apprentissage+pour+reseau+a+seuil+assymetrique%2E+Cognitiva+85%3A+A+la+Frontiere+de+l%27Intelligence+Artificielle+des+Sciences+de+la+Connaissance+des+Neurosciences%2C+599%2D604)
9. LeCun, Y., Boser, B., Denker, J.S., Henderson, D., Howard, R.E., Hubbard, W., & Jackel, L.D. (1990). Handwritten digit recognition with a back-propagation network. Advances in Neural Information Processing Systems, 2, 396-404, Morgan Kaufman. [find](https://scholar.google.com/scholar?q=LeCun%2C+Y%2E%2C+Boser%2C+B%2E%2C+Denker%2C+J%2ES%2E%2C+Henderson%2C+D%2E%2C+Howard%2C+R%2EE%2E%2C+Hubbard%2C+W%2E%2C+%26+Jackel%2C+L%2ED%2E+%281990%29%2E+Handwritten+digit+recognition+with+a+back%2Dpropagation+network%2E+Advances+in+Neural+Information+Proce)
10. Parker, D.B. (1985). Learning logic. Technical Report TR-47, Center for Computational Research in Economics and Management Science, Massachusetts Institute of Technology, Cambridge, MA. [find](https://scholar.google.com/scholar?q=Parker%2C+D%2EB%2E+%281985%29%2E+Learning+logic%2E+Technical+Report+TR%2D47%2C+Center+for+Computational+Research+in+Economics+and+Management+Science%2C+Massachusetts+Institute+of+Technology%2C+Cambridge%2C+MA%2E)
11. Rosenblatt, F. (1962). Principles ofNeurodynamics, Spartan Books, New York. ~i ä 1 [ A r D A + I A r A ] [find](https://scholar.google.com/scholar?q=Rosenblatt%2C+F%2E+%281962%29%2E+Principles+ofNeurodynamics%2C+Spartan+Books%2C+New+York%2E+%7Ei+%C3%A4+1+%5B+A+r+D+A+%2B+I+A+r+A+%5D)
12. (67) SUPPORT-VECTORNETWORKS Rumelhart, D.E., Hinton, G.E., & Williams, R.J. (1986). Learning intemal representations by backpropagating errors. Nature, 323:533-536. [find](https://scholar.google.com/scholar?q=%2867%29+SUPPORT%2DVECTORNETWORKS+Rumelhart%2C+D%2EE%2E%2C+Hinton%2C+G%2EE%2E%2C+%26+Williams%2C+R%2EJ%2E+%281986%29%2E+Learning+intemal+representations+by+backpropagating+errors%2E+Nature%2C+323%3A533%2D536%2E)
13. Rumelhart, D.E., Hinton, G.E., & Williams, R.J. (1987). Learning intemal representations by error propagation. In James L. McClelland & David E. Rumelhart (Eds.), Parallel Distributed Processing, 1, 318-362, MIT Press. [find](https://scholar.google.com/scholar?q=Rumelhart%2C+D%2EE%2E%2C+Hinton%2C+G%2EE%2E%2C+%26+Williams%2C+R%2EJ%2E+%281987%29%2E+Learning+intemal+representations+by+error+propagation%2E+In+James+L%2E+McClelland+%26+David+E%2E+Rumelhart+%28Eds%2E%29%2C+Parallel+Distributed+Processing%2C+1%2C+3)
14. Vapnik, V.N. (1982). Estimation ofDependences Based on Empirical Data, Addendum 1, New York: Springer-Verlag. [find](https://scholar.google.com/scholar?q=Vapnik%2C+V%2EN%2E+%281982%29%2E+Estimation+ofDependences+Based+on+Empirical+Data%2C+Addendum+1%2C+New+York%3A+Springer%2DVerlag%2E)
15. Received May 15, 1993 Accepted February 20, 1995 Final Manuscript March 8, 1995 [find](https://scholar.google.com/scholar?q=Received+May+15%2C+1993+Accepted+February+20%2C+1995+Final+Manuscript+March+8%2C+1995)

*Updated 29 October 2026*