## The Annals of Probability

### Large deviations for random projections of $\ell^{p}$ balls

#### Abstract

Let $p\in[1,\infty]$. Consider the projection of a uniform random vector from a suitably normalized $\ell^{p}$ ball in $\mathbb{R}^{n}$ onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimension $n$ goes to $\infty$, which can be viewed as an annealed LDP. We also establish a quenched LDP (conditioned on a fixed sequence of projection directions) and show that for $p\in(1,\infty]$ (but not for $p=1$), the corresponding rate function is “universal,” in the sense that it coincides for “almost every” sequence of projection directions. We also analyze some exceptional sequences of directions in the “measure zero” set, including the sequence of directions corresponding to the classical Cramér’s theorem, and show that those sequences of directions yield LDPs with rate functions that are distinct from the universal rate function of the quenched LDP. Lastly, we identify a variational formula that relates the annealed and quenched LDPs, and analyze the minimizer of this variational formula. These large deviation results complement the central limit theorem for convex sets, specialized to the case of sequences of $\ell^{p}$ balls.

#### Article information

Source
Ann. Probab., Volume 45, Number 6B (2017), 4419-4476.

Dates
Revised: November 2016
First available in Project Euclid: 12 December 2017

https://projecteuclid.org/euclid.aop/1513069264

Digital Object Identifier
doi:10.1214/16-AOP1169

Mathematical Reviews number (MathSciNet)
MR3737915

Zentralblatt MATH identifier
06838124

#### Citation

Gantert, Nina; Kim, Steven Soojin; Ramanan, Kavita. Large deviations for random projections of $\ell^{p}$ balls. Ann. Probab. 45 (2017), no. 6B, 4419--4476. doi:10.1214/16-AOP1169. https://projecteuclid.org/euclid.aop/1513069264

#### References

• [1] Aidékon, E. (2010). Large deviations for transient random walks in random environment on a Galton–Watson tree. Ann. Inst. Henri Poincaré Probab. Stat. 46 159–189.
• [2] Anttila, M., Ball, K. and Perissinaki, I. (2003). The central limit problem for convex bodies. Trans. Amer. Math. Soc. 355 4723–4735.
• [3] Arcones, M. A. (2002). Large and moderate deviations of empirical processes with nonstandard rates. Statist. Probab. Lett. 57 315–326.
• [4] Ball, K. (1988). Logarithmically concave functions and sections of convex sets in $\mathbf{R}^{n}$. Studia Math. 88 69–84.
• [5] Barthe, F., Gamboa, F., Lozada-Chang, L.-V. and Rouault, A. (2010). Generalized Dirichlet distributions on the ball and moments. ALEA Lat. Am. J. Probab. Math. Stat. 7 319–340.
• [6] Barthe, F., Guédon, O., Mendelson, S. and Naor, A. (2005). A probabilistic approach to the geometry of the $l^{n}_{p}$-ball. Ann. Probab. 33 480–513.
• [7] Barthe, F. and Koldobsky, A. (2003). Extremal slabs in the cube and the Laplace transform. Adv. Math. 174 89–114.
• [8] Barthe, F. and Naor, A. (2002). Hyperplane projections of the unit ball of $\ell_{p}^{n}$. Discrete Comput. Geom. 27 215–226.
• [9] Ben Arous, G., Dembo, A. and Guionnet, A. (2001). Aging of spherical spin glasses. Probab. Theory Related Fields 120 1–67.
• [10] Bercu, B., Gassiat, E. and Rio, E. (2002). Concentration inequalities, large and moderate deviations for self-normalized empirical processes. Ann. Probab. 30 1576–1604.
• [11] Bingham, E. and Mannila, H. (2001). Random projection in dimensionality reduction: Applications to image and text data. In Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining 245–250. ACM, New York.
• [12] Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge Univ. Press, Cambridge.
• [13] Brehm, U. and Voigt, J. (2000). Asymptotics of cross sections for convex bodies. Beitr. Algebra Geom. 41 437–454.
• [14] Comets, F., Gantert, N. and Zeitouni, O. (2000). Quenched, annealed and functional large deviations for one-dimensional random walk in random environment. Probab. Theory Related Fields 118 65–114.
• [15] Cramér, H. (1938). Sur un nouveau théoréme–limite de la théorie des probabilités. Actual. Sci. Ind. 736 5–23.
• [16] de Acosta, A. (1985). On large deviations of sums of independent random vectors. In Probability in Banach Spaces, V (Medford, Mass., 1984). Lecture Notes in Math. 1153 1–14. Springer, Berlin.
• [17] Dembo, A. and Shao, Q.-M. (1998). Self-normalized large deviations in vector spaces. In High Dimensional Probability (Oberwolfach, 1996). Progress in Probability 43 27–32. Birkhäuser, Basel.
• [18] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. Springer, New York.
• [19] de la Peña, V. H., Lai, T. L. and Shao, Q.-M. (2009). Self-Normalized Processes: Limit Theory and Statistical Applications. Springer, Berlin.
• [20] Diaconis, P. and Freedman, D. (1984). Asymptotics of graphical projection pursuit. Ann. Statist. 12 793–815.
• [21] Ding, W., Rohban, M. H., Ishwar, P. and Saligrama, V. (2013). Topic discovery through data dependent and random projections. J. Mach. Learn. Res. 28. To appear.
• [22] Eichelsbacher, P. and Schmock, U. (1998). Exponential approximations in completely regular topological spaces and extensions of Sanov’s theorem. Stochastic Process. Appl. 77 233–251.
• [23] Fern, X. Z. and Brodley, C. E. (2003). Random projection for high dimensional data clustering: A cluster ensemble approach. In ICML 3 186–193.
• [24] Fleury, B., Guédon, O. and Paouris, G. (2007). A stability result for mean width of $L_{p}$-centroid bodies. Adv. Math. 214 865–877.
• [25] Foss, S., Konstantopoulos, T. and Zachary, S. (2007). Discrete and continuous time modulated random walks with heavy-tailed increments. J. Theoret. Probab. 20 581–612.
• [26] Gantert, N. and Kim, S. S. and Ramanan, K. (2015). Large deviations for random projections of $\ell^{p}$ balls. Preprint. Available at arXiv:1512.04988.
• [27] Gantert, N., Kim, S. S. and Ramanan, K. (2016). Cramér’s theorem is atypical. In Advances in the Mathematical Sciences: Research from the 2015 Association for Women in Mathematics Symposium (G. Letzter, K. Lauter, E. Chambers, N. Flournoy, J. E. Grigsby, C. Martin, K. Ryan and K. Trivisa, eds.) 253–270. Springer International Publishing, Cham.
• [28] Gantert, N., Ramanan, K. and Rembart, F. (2014). Large deviations for weighted sums of stretched exponential random variables. Electron. Commun. Probab. 19 1–14.
• [29] Gnedenko, B. (1943). Sur la distribution limite du terme maximum d’une série aléatoire. Ann. of Math. (2) 44 423–453.
• [30] Indyk, P. (2000). Stable distributions, pseudorandom generators, embeddings and data stream computation. In 41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA, 2000) 189–197. IEEE Comput. Soc. Press, Los Alamitos, CA.
• [31] Jiang, T. (2005). Maxima of entries of Haar distributed matrices. Probab. Theory Related Fields 131 121–144.
• [32] Jing, B.-Y., Shao, Q.-M., Wang, Q. et al. (2003). Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab. 31 2167–2215.
• [33] Kim, S. S. and Ramanan, K. (2015). A Sanov-type theorem for empirical measures associated with the surface and cone measures on $\ell^{p}$ spheres. Preprint. Available at arXiv:1509.05442.
• [34] Klartag, B. (2007). A central limit theorem for convex sets. Invent. Math. 168 91–131.
• [35] Ledoux, M. (1996). Isoperimetry and Gaussian analysis. In Lectures on Probability Theory and Statistics (Saint-Flour, 1994). Lecture Notes in Math. 1648 165–294. Springer, Berlin.
• [36] Lin, J. and Gunopulos, D. (2003). Dimensionality reduction by random projection and latent semantic indexing. In Proceedings of the Text Mining Workshop, at the 3rd SIAM International Conference on Data Mining Citeseer.
• [37] Maillard, O.-A. and Munos, R. (2012). Linear regression with random projections. J. Mach. Learn. Res. 13 2735–2772.
• [38] Meckes, E. (2012). Approximation of projections of random vectors. J. Theoret. Probab. 25 333–352.
• [39] Meckes, E. (2012). Projections of probability distributions: A measure-theoretic Dvoretzky theorem. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 2050 317–326. Springer, Heidelberg.
• [40] Meyer, M. and Pajor, A. (1988). Sections of the unit ball of $L^{n}_{p}$. J. Funct. Anal. 80 109–123.
• [41] Mikosch, T. and Nagaev, A. (1998). Large deviations of heavy-tailed sums with applications in insurance. Extremes 1 81–110.
• [42] Naor, A. (2007). The surface measure and cone measure on the sphere of $\ell_{p}^{n}$. Trans. Amer. Math. Soc. 359 1045–1079.
• [43] Naor, A. and Romik, D. (2003). Projecting the surface measure of the sphere of $\ell_{p}^{n}$. Ann. Inst. Henri Poincaré Probab. Stat. 39 241–261.
• [44] Patterson, R. F. and Taylor, R. L. (1985). Strong laws of large numbers for triangular arrays of exchangeable random variables. Stoch. Anal. Appl. 3 171–187.
• [45] Rachev, S. T. and Rüschendorf, L. (1991). Approximate independence of distributions on spheres and their stability properties. Ann. Probab. 19 1311–1337.
• [46] Resnick, S. I. and Tomkins, R. J. (1973). Almost sure stability of maxima. J. Appl. Probab. 10 387–401.
• [47] Rockafellar, R. T. (1970). Convex Analysis 28. Princeton Univ. Press, Princeton.
• [48] Romano, J. P. and Siegel, A. F. (1986). Counterexamples in Probability and Statistics. CRC Press, Boca Raton.
• [49] Schechtman, G. and Zinn, J. (1990). On the volume of the intersection of two $L^{n}_{p}$ balls. Proc. Amer. Math. Soc. 110 217–224.
• [50] Shao, Q.-M. (1997). Self-normalized large deviations. Ann. Probab. 25 285–328.
• [51] Shorack, G. R. and Wellner, J. A. (2009). Empirical Processes with Applications to Statistics. Classics in Applied Mathematics 59. SIAM, Philadelphia, PA.
• [52] Sion, M. (1958). On general minimax theorems. Pacific J. Math. 8 171–176.
• [53] Sodin, S. (2007). Tail-sensitive Gaussian asymptotics for marginals of concentrated measures in high dimension. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1910 271–295. Springer, Berlin.
• [54] Spruill, M. C. (2007). Asymptotic distribution of coordinates on high dimensional spheres. Electron. Commun. Probab. 12 234–247.
• [55] Sudakov, V. N. (1978). Typical distributions of linear functionals in finite-dimensional spaces of high dimension. Dokl. Akad. Nauk SSSR 243 1402–1405.
• [56] Trashorras, J. (2002). Large deviations for a triangular array of exchangeable random variables. Ann. Inst. Henri Poincaré Probab. Stat. 38 649–680.
• [57] Villani, C. (2009). Optimal Transport: Old and New. Springer, Berlin.