An augmented Lagrangian approach for sparse principal component analysis

Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components (PCs) are usually linear combinations of all the original variables, and it is thus often difficult to interpret the PCs. To alleviate this drawback, various sparse PCA approaches were proposed in the literature (Cadima and Jolliffe in J Appl Stat 22:203–214, 1995; d’Aspremont et?al. in J Mach Learn Res 9:1269–1294, 2008; d’Aspremont et?al. SIAM Rev 49:434–448, 2007; Jolliffe in J Appl Stat 22:29–35, 1995; Journée et?al. in J Mach Learn Res 11:517–553, 2010; Jolliffe et?al. in J Comput Graph Stat 12:531–547, 2003; Moghaddam et?al. in Advances in neural information processing systems 18:915–922, MIT Press, Cambridge, 2006; Shen and Huang in J Multivar Anal 99(6):1015–1034, 2008; Zou et?al. in J Comput Graph Stat 15(2):265–286, 2006). Despite success in achieving sparsity, some important properties enjoyed by the standard PCA are lost in these methods such as uncorrelation of PCs and orthogonality of loading vectors. Also, the total explained variance that they attempt to maximize can be too optimistic. In this paper we propose a new formulation for sparse PCA, aiming at finding sparse and nearly uncorrelated PCs with orthogonal loading vectors while explaining as much of the total variance as possible. We also develop a novel augmented Lagrangian method for solving a class of nonsmooth constrained optimization problems, which is well suited for our formulation of sparse PCA. We show that it converges to a feasible point, and moreover under some regularity assumptions, it converges to a stationary point. Additionally, we propose two nonmonotone gradient methods for solving the augmented Lagrangian subproblems, and establish their global and local convergence. Finally, we compare our sparse PCA approach with several existing methods on synthetic (Zou et?al. in J Comput Graph Stat 15(2):265–286, 2006), Pitprops (Jeffers in Appl Stat 16:225–236, 1967), and gene expression data (Chin et?al in Cancer Cell 10:529C–541C, 2006), respectively. The computational results demonstrate that the sparse PCs produced by our approach substantially outperform those by other methods in terms of total explained variance, correlation of PCs, and orthogonality of loading vectors. Moreover, the experiments on random data show that our method is capable of solving large-scale problems within a reasonable amount of time.     

3-Nets realizing a group in a projective plane

In a projective plane $\mathit{PG}(2,\mathbb{K})$ defined over an algebraically closed field $\mathbb{K}$ of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614–1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672–688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzúa’s 3-nets (Adv. Geom. 10:287–310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds in characteristic p>0 apart from three possible exceptions $\rm{Alt}_{4}$ , $\rm{Sym}_{4}$ , and $\rm{Alt}_{5}$ . Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069–1088, 2007; Miguel and Buzunáriz in Graphs Comb. 25:469–488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672–688, 2008; Yuzvinsky in Compos. Math. 140:1614–1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641–1648, 2009).     

Newton Polyhedra of Discriminants of Projections

For a system of polynomial equations, whose coefficients depend on parameters, the Newton polyhedron of its discriminant is computed in terms of the Newton polyhedra of the coefficients. This leads to an explicit formula (involving Euler obstructions of toric varieties) in the unmixed case, suggests certain open questions in general, and generalizes a number of similar known results (Gelfand et al. in Discriminants, resultants, and multidimensional determinants. Birkhäuser, Boston, 1994; Sturmfels in J. Algebraic Comb. 32(2):207–236, 1994; McDonald in Discrete Comput. Geom. 27:501–529, 2002; Gonzalez-Perez in Can. J. Math. 52(2):348-368, 2000; Esterov and Khovanskii in Funct. Anal. Math. 2(1), 2008).     

More about λ-support iterations of (<λ)-complete forcing notions

This article continues Ros?anowski and Shelah (Int J Math Math Sci 28:63–82, 2001; Quaderni di Matematica 17:195–239, 2006; Israel J Math 159:109–174, 2007; 2011; Notre Dame J Formal Logic 52:113–147, 2011) and we introduce here a new property of (<λ)-strategically complete forcing notions which implies that their λ-support iterations do not collapse λ ⁺ (for a strongly inaccessible cardinal λ).     

An improved local convergence analysis for a two-step Steffensen-type method

We study a class of Steffensen-type algorithm for solving nonsmooth variational inclusions in Banach spaces. We provide a local convergence analysis under ω-conditioned divided difference, and the Aubin continuity property. This work on the one hand extends the results on local convergence of Steffensen’s method related to the resolution of nonlinear equations (see Amat and Busquier in Comput. Math. Appl. 49:13–22, 2005; J. Math. Anal. Appl. 324:1084–1092, 2006; Argyros in Southwest J. Pure Appl. Math. 1:23–29, 1997; Nonlinear Anal. 62:179–194, 2005; J. Math. Anal. Appl. 322:146–157, 2006; Rev. Colomb. Math. 40:65–73, 2006; Computational Theory of Iterative Methods, 2007). On the other hand our approach improves the ratio of convergence and enlarges the convergence ball under weaker hypotheses than one given in Hilout (Commun. Appl. Nonlinear Anal. 14:27–34, 2007).     

Strong NP-hardness of scheduling problems with learning or aging effect

Proofs of strong NP-hardness of single machine and two-machine flowshop scheduling problems with learning or aging effect given in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c; Applied Mathematical Modelling 37:1523–1536, 2013) contain a common mistake that make them incomplete. We reveal the mistake and provide necessary corrections for the problems in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; Applied Mathematical Modelling 37:1523–1536, 2013). NP-hardness of problems in Rudek (International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c) remains unknown because of another mistake which we are unable to correct.     

On rigidity of hypersurfaces with constant curvature functions in warped product manifolds

In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as “weighted” mean curvatures, which extend the work (Brendle in Publ Math Inst Hautes Études Sci 117:247–269, 2013; Brendle and Eichmair in J Differ Geom 94(94):387–407, 2013; Montiel in Indiana Univ Math J 48:711–748, 1999) considering constant mean curvature functions. Secondly, we obtain the rigidity results for hypersurfaces in the space forms with constant linear combinations of intrinsic Gauss–Bonnet curvatures $L_k$ . To achieve this, we develop some new kind of Newton–Maclaurin type inequalities on $L_k$ which may have independent interest.     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

Index sets and Scott sentences

For a computable structure \({\mathcal{A}}\) , there may not be a computable infinitary Scott sentence. When there is a computable infinitary Scott sentence \({\varphi}\) , then the complexity of the index set \({I(\mathcal{A})}\) is bounded by that of \({\varphi}\) . There are results (Ash and Knight in Computable structures and the hyperarithmetical hierarchy. Elsevier, Amsterdam, 2000; Calvert et al. in Algeb Log 45:306–315, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012; Knight and Saraph in Scott sentences for certain groups, pre-print) giving “optimal” Scott sentences for structures of various familiar kinds. These results have been driven by the thesis that the complexity of the index set should match that of an optimal Scott sentence (Ash and Knight in Computable structures and the hyperarithmetical hierarchy. Elsevier, Amsterdam, 2000; Calvert et al. in Algeb Log 45:306–315, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012). In this note, it is shown that the thesis does not always hold. For a certain subgroup of \({\mathbb{Q}}\) , there is no computable d- \({\Sigma_2}\) Scott sentence, even though (as shown in Ash and Knight in Scott sentences for certain groups, pre-print) the index set is d- \({\Sigma^0_2}\) .     

Constructive finite free resolutions

Northcott’s book Finite Free Resolutions (1976), as well as the paper (J. Reine Angew. Math. 262/263:205–219, 1973), present some key results of Buchsbaum and Eisenbud (J. Algebra 25:259–268, 1973; Adv. Math. 12: 84–139, 1974) both in a simplified way and without Noetherian hypotheses, using the notion of latent nonzero divisor introduced by Hochster. The goal of this paper is to simplify further the proofs of these results, which become now elementary in a logical sense (no use of prime ideals, or minimal prime ideals) and, we hope, more perspicuous. Some formulations are new and more general than in the references (J. Algebra 25:259–268, 1973; Adv. Math. 12: 84–139, 1974; Finite Free Resolutions 1976) (Theorem 7.2, Lemma 8.2 and Corollary 8.5).     

Parabolic constructions of asymptotically flat 3-metrics of prescribed scalar curvature

In 1993, Bartnik (J. Differ. Geom. 37(1):37–71) introduced a quasi-spherical construction of metrics of prescribed scalar curvature on 3-manifolds. Under quasi-spherical ansatz, the problem is converted into the initial value problem for a semi-linear parabolic equation of the lapse function. The original ansatz of Bartnik started with a background foliation with round metrics on the 2-sphere leaves. This has been generalized by several authors (Shi and Tam in J. Differ. Geom. 62(1):79–125, 2002; Smith in Gen. Relat. Gravit. 41(5):1013–1024, 2009; Smith and Weinstein in Commun. Anal. Geom. 12(3):511–551, 2004) under various assumptions on the background foliation. In this article, we consider background foliations given by conformal round metrics, and by the Ricci flow on 2-spheres. We discuss conditions on the scalar curvature function and on the foliation that guarantee the solvability of the parabolic equation, and thus the existence of asymptotically flat 3-metrics with a prescribed inner boundary. In particular, many examples of asymptotically flat-scalar flat 3-metrics with outermost minimal surfaces are obtained.     

Double Bubbles for Immiscible Fluids in ℝ n

We use a new approach that we call unification to prove that standard weighted double bubbles in n-dimensional Euclidean space minimize immiscible fluid surface energy, that is, surface area weighted by constants. The result is new for weighted area, and also gives the simplest known proof to date of the (unit weight) double bubble theorem (Hass et al., Electron. Res. Announc. Am. Math. Soc., 1(3):98–102, 1995; Hutchings et al., Ann. Math., 155(2):459–489, 2002; Reichardt, J. Geom. Anal., 18(1):172–191, 2008). As part of the proof, we introduce a striking new symmetry argument for showing that a minimizer must be a surface of revolution.     

On Ulm’s method for Fréchet differentiable operators

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Izv. Akad. Nauk Est. SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) do not. We also show that under the same hypotheses and computational cost as (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) finer error sequences can be obtained. Numerical examples are also provided further validating the results.     

Non-Decaying Solutions to the Navier Stokes Equations in Exterior Domains

We establish a new theorem of existence (and uniqueness) of solutions to the Navier-Stokes initial boundary value problem in exterior domains. No requirement is made on the convergence at infinity of the kinetic field and of the pressure field. These solutions are called non-decaying solutions. The first results on this topic dates back about 40 years ago see the references (Galdi and Rionero in Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980; Knightly in SIAM J. Math. Anal. 3:506–511, 1972). In the articles Galdi and Rionero (Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980) it was introduced the so called weight function method to study the uniqueness of solutions. More recently, the problem has been considered again by several authors (see Galdi et al. in J. Math. Fluid Mech. 14:633–652, 2012, Quad. Mat. 4:27–68, 1999, Nonlinear Anal. 47:4151–4156, 2001; Kato in Arch. Ration. Mech. Anal. 169:159–175, 2003; Kukavica and Vicol in J. Dyn. Differ. Equ. 20:719–732, 2008; Maremonti in Mat. Ves. 61:81–91, 2009, Appl. Anal. 90:125–139, 2011).     

Classroom-based interventions in mathematics education: relevance, significance, and applicability

This special issue discusses various pedagogical innovations and myriad of significant findings. This commentary is not a synthesis of these contributions, but a summary of my own reflections on selected aspects of the nine papers comprising the special issue. Four themes subsume these reflections: (1) Gestural Communication (Alibali, Nathan, Church, Wolfgram, Kim and Knuth 2013); (2) Development of Ways of Thinking (Jahnke and Wambach 2013; Lehrer, Kobiela and Weinberg 2013; Mariotti 2013; Roberts and A. Stylianides 2013; Shilling-Traina and G. Stylianides 2013; Tabach, Hershkowitz and Dreyfus 2013); (3) Learning Mathematics through Representation (Saxe, Diakow and Gearhart 2013); and (4) Challenges in Dialogic Teaching (Ruthven and Hofmann 2013).     

Characterization of the Fermat curve as the most symmetric nonsingular algebraic plane curve

A projective nonsingular plane algebraic curve of degree \(d\ge 4\) is called maximally symmetric if it attains the maximum order of the automorphism groups for complex nonsingular plane algebraic curves of degree \(d\) . For \(d\le 7\) , all such curves are known. Up to projectivities, they are the Fermat curve for \(d=5,7\) ; see Kaneta et al. (RIMS Kokyuroku 1109:182–191, 1999) and Kaneta et al. (Geom. Dedic. 85:317–334, 2001), the Klein quartic for \(d=4\) , see Hartshorne (Algebraic Geometry. Springer, New York, 1977), and the Wiman sextic for \(d=6\) ; see Doi et al. (Osaka J. Math. 37:667–687, 2000). In this paper we work on projective plane curves defined over an algebraically closed field of characteristic zero, and we extend this result to every \(d\ge 8\) showing that the Fermat curve is the unique maximally symmetric nonsingular curve of degree \(d\) with \(d\ge 8\) , up to projectivity. For \(d=11,13,17,19\) , this characterization of the Fermat curve has already been obtained; see Kaneta et al. (Geom. Dedic. 85:317–334, 2001).     

The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups

The general surface group conjecture asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related conjectures given in Fine et al. (Sci Math A 1:1–15, 2008). First we obtain the Surface Group Conjecture B for cyclically pinched and conjugacy pinched one-relator groups. That is: if G is a cyclically pinched one-relator group or conjugacy pinched one-relator group satisfying property IF then G is free, a surface group or a solvable Baumslag–Solitar Group. Further combining results in Fine et al. (Sci Math A 1:1–15, 2008) on Property IF with a theorem of Wilton (Geom Topol, 2012) and results of Stallings (Ann Math 2(88):312–334, 1968) and Kharlampovich and Myasnikov (Trans Am Math Soc 350(2):571–613, 1998) we show that Surface Group Conjecture C proposed in Fine et al. (Sci Math A 1:1–15, 2008) is true, namely: If G is a finitely generated nonfree freely indecomposable fully residually free group with property IF, then G is a surface group.     

The Problem of Massive Deception for Justification Norms of Action

In this paper, I argue against recent versions of justification norms of action and practical deliberation (Neta, Noûs 43:684–699, 2009; Gerken, Synthese 178:529–547, 2011, Synthese 189:373–394, 2012; Smithies, Noûs 46:265–288, 2012). I demonstrate that these norms yield unacceptable results in deception cases. However, a further modification of justification norms in the light of these results appears to be ad hoc. Hence, I claim, we should reject justification norms of action and practical deliberation.     

Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq Equations

We establish a connection between optimal transport theory (see Villani in Topics in optimal transportation. Graduate studies in mathematics, vol. 58, AMS, Providence, 2003, for instance) and classical convection theory for geophysical flows (Pedlosky, in Geophysical fluid dynamics, Springer, New York, 1979). Our starting point is the model designed few years ago by Angenent, Haker, and Tannenbaum (SIAM J. Math. Anal. 35:61–97, 2003) to solve some optimal transport problems. This model can be seen as a generalization of the Darcy–Boussinesq equations, which is a degenerate version of the Navier–Stokes–Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized hydrostatic-Boussinesq equations) to various models involving optimal transport (and the related Monge–Ampère equation, Brenier in Commun. Pure Appl. Math. 64:375–417, 1991; Caffarelli in Commun. Pure Appl. Math. 45:1141–1151, 1992). This includes the 2D semi-geostrophic equations (Hoskins in Annual review of fluid mechanics, vol. 14, pp. 131–151, Palo Alto, 1982; Cullen et al. in SIAM J. Appl. Math. 51:20–31, 1991, Arch. Ration. Mech. Anal. 185:341–363, 2007; Benamou and Brenier in SIAM J. Appl. Math. 58:1450–1461, 1998; Loeper in SIAM J. Math. Anal. 38:795–823, 2006) and some fully nonlinear versions of the so-called high-field limit of the Vlasov–Poisson system (Nieto et al. in Arch. Ration. Mech. Anal. 158:29–59, 2001) and of the Keller–Segel for Chemotaxis (Keller and Segel in J. Theor. Biol. 30:225–234, 1971; Jäger and Luckhaus in Trans. Am. Math. Soc. 329:819–824, 1992; Chalub et al. in Mon. Math. 142:123–141, 2004). Mathematically speaking, we establish some existence theorems for local smooth, global smooth or global weak solutions of the different models. We also justify that the inertia terms can be rigorously neglected under appropriate scaling assumptions in the generalized Navier–Stokes–Boussinesq equations. Finally, we show how a “stringy” generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology (see Arnold and Khesin in Topological methods in hydrodynamics. Applied mathematical sciences, vol. 125, Springer, Berlin, 1998; Moffatt in J. Fluid Mech. 159:359–378, 1985, Topological aspects of the dynamics of fluids and plasmas. NATO adv. sci. inst. ser. E, appl. sci., vol. 218, Kluwer, Dordrecht, 1992; Schonbek in Theory of the Navier–Stokes equations, Ser. adv. math. appl. sci., vol. 47, pp. 179–184, World Sci., Singapore, 1998; Vladimirov et al. in J. Fluid Mech. 390:127–150, 1999; Nishiyama in Bull. Inst. Math. Acad. Sin. (N.S.) 2:139–154, 2007).     

Universal Lagrangian bundles

The obstruction to construct a Lagrangian bundle over a fixed integral affine manifold was constructed by Dazord and Delzant (J Differ Geom 26:223–251, 1987) and shown to be given by ‘twisted’ cup products in Sepe (Differ Geom Appl 29(6): 787–800, 2011). This paper uses the topology of universal Lagrangian bundles, which classify Lagrangian bundles topologically [cf. Sepe in J Geom Phys 60:341–351, 2010], to reinterpret this obstruction as the vanishing of a differential on the second page of a Leray-Serre spectral sequence. Using this interpretation, it is shown that the obstruction of Dazord and Delzant depends on an important cohomological invariant of the integral affine structure on the base space, called the radiance obstruction, which was introduced by Goldman and Hirsch (Trans Am Math Soc 286(2):629–649, 1984). Some examples, related to non-degenerate singularities of completely integrable Hamiltonian systems, are discussed.