Quantum Query Complexity of Almost All Functions with Fixed On-set Size

This paper considers the quantum query complexity of almost all functions in the set \({\mathcal{F}}_{N,M}\) of \({N}\)-variable Boolean functions with on-set size \({M (1\le M \le 2^{N}/2)}\), where the on-set size is the number of inputs on which the function is true. The main result is that, for all functions in \({\mathcal{F}}_{N,M}\) except its polynomially small fraction, the quantum query complexity is \({ \Theta\left(\frac{\log{M}}{c + \log{N} - \log\log{M}} + \sqrt{N}\right)}\) for a constant \({c > 0}\). This is quite different from the quantum query complexity of the hardest function in \({\mathcal{F}}_{N,M}\): \({\Theta\left(\sqrt{N\frac{\log{M}}{c + \log{N} - \log\log{M}}} + \sqrt{N}\right)}\). In contrast, almost all functions in \({\mathcal{F}}_{N,M}\) have the same randomized query complexity \({\Theta(N)}\) as the hardest one, up to a constant factor.     

Testing list <Emphasis Type="Italic">H</Emphasis>-homomorphisms

In the List H- Homomorphism Problem, for a graph H that is a parameter of the problem, an instance consists of an undirected graph G with a list constraint \({L(v) \subseteq V(H)}\) for each variable \({v \in V(G)}\), and the objective is to determine whether there is a list H-homomorphism \({f:V(G) \to V(H)}\), that is, \({f(v) \in L(v)}\) for every \({v \in V(G)}\) and \({(f(u),f(v)) \in E(H)}\) whenever \({(u,v) \in E(G)}\).We consider the problem of testing list H-homomorphisms in the following weighted setting: An instance consists of an undirected graph G, list constraints L, weights imposed on the vertices of G, and a map \({f:V(G) \to V(H)}\) given as an oracle access. The objective is to determine whether f is a list H-homomorphism or far from any list H-homomorphism. The farness is measured by the total weight of vertices \({v \in V(G)}\) for which f(v) must be changed so as to make f a list H-homomorphism. In this paper, we classify graphs H with respect to the number of queries to f required to test the list H-homomorphisms. Specifically, we show that (i) list H-homomorphisms are testable with a constant number of queries if and only if H is a reflexive complete graph or an irreflexive complete bipartite graph and (ii) list H-homomorphisms are testable with a sublinear number of queries if and only if H is a bi-arc graph.     

Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

In this paper, we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds, we obtain lower bounds for these models.For depth-3 multilinear formulas, of size exp\({(n^\delta)}\), we give a hitting set of size exp\({\left(\tilde{O}\left(n^{2/3 + 2\delta/3}\right) \right)}\). This implies a lower bound of exp\({(\tilde{\Omega}(n^{1/2}))}\) for depth-3 multilinear formulas, for some explicit polynomial.For depth-4 multilinear formulas, of size exp\({(n^\delta)}\), we give a hitting set of size exp\({\left(\tilde{O}\left(n^{2/3 + 4\delta/3}\right) \right)}\). This implies a lower bound of exp\({(\tilde{\Omega}(n^{1/4}))}\) for depth-4 multilinear formulas, for some explicit polynomial.A regular formula consists of alternating layers of \({+,\times}\) gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp\({\left(n^{1- \delta}\right)}\), for regular depth-d multilinear formulas with formal degree at most n and size exp\({(n^\delta)}\), where \({\delta = O(1/{\sqrt{5}^d})}\). This result implies a lower bound of roughly exp\({(\tilde{\Omega}(n^{1/{\sqrt{5}^d}}))}\) for such formulas.We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known.Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a read-once algebraic branching program (from depth-3 formulas, we go straight to read-once algebraic branching programs).     

Incompressible Functions,Relative-Error Extractors,and the Power of Nondeterministic Reductions

A circuit C compresses a function \({f : \{0,1\}^n\rightarrow \{0,1\}^m}\) if given an input \({x\in \{0,1\}^n}\), the circuit C can shrink x to a shorter ?-bit string x′ such that later, a computationally unbounded solver D will be able to compute f(x) based on x′. In this paper we study the existence of functions which are incompressible by circuits of some fixed polynomial size \({s=n^c}\). Motivated by cryptographic applications, we focus on average-case \({(\ell,\epsilon)}\) incompressibility, which guarantees that on a random input \({x\in \{0,1\}^n}\), for every size s circuit \({C:\{0,1\}^n\rightarrow \{0,1\}^{\ell}}\) and any unbounded solver D, the success probability \({\Pr_x[D(C(x))=f(x)]}\) is upper-bounded by \({2^{-m}+\epsilon}\). While this notion of incompressibility appeared in several works (e.g., Dubrov and Ishai, STOC 06), so far no explicit constructions of efficiently computable incompressible functions were known. In this work, we present the following results:

1. (1)
   Assuming that E is hard for exponential size nondeterministic circuits, we construct a polynomial time computable boolean function \({f:\{0,1\}^n\rightarrow \{0,1\}}\) which is incompressible by size n ^c circuits with communication \({\ell=(1-o(1)) \cdot n}\) and error \({\epsilon=n^{-c}}\). Our technique generalizes to the case of PRGs against nonboolean circuits, improving and simplifying the previous construction of Shaltiel and Artemenko (STOC 14).
    
2. (2)
   We show that it is possible to achieve negligible error parameter \({\epsilon=n^{-\omega(1)}}\) for nonboolean functions. Specifically, assuming that E is hard for exponential size \({\Sigma_3}\)-circuits, we construct a nonboolean function \({f:\{0,1\}^n\rightarrow \{0,1\}^m}\) which is incompressible by size n ^c circuits with \({\ell=\Omega(n)}\) and extremely small \({\epsilon=n^{-c} \cdot 2^{-m}}\). Our construction combines the techniques of Trevisan and Vadhan (FOCS 00) with a new notion of relative error deterministic extractor which may be of independent interest.
    
3. (3)
   We show that the task of constructing an incompressible boolean function \({f:\{0,1\}^n\rightarrow \{0,1\}}\) with negligible error parameter \({\epsilon}\) cannot be achieved by “existing proof techniques”. Namely, nondeterministic reductions (or even \({\Sigma_i}\) reductions) cannot get \({\epsilon=n^{-\omega(1)}}\) for boolean incompressible functions. Our results also apply to constructions of standard Nisan-Wigderson type PRGs and (standard) boolean functions that are hard on average, explaining, in retrospect, the limitations of existing constructions. Our impossibility result builds on an approach of Shaltiel and Viola (STOC 08).
    

     

Lower Bounds for Depth-Three Arithmetic Circuits with small bottom fanin

Shpilka & Wigderson (IEEE conference on computational complexity, vol 87, 1999) had posed the problem of proving exponential lower bounds for (nonhomogeneous) depth-three arithmetic circuits with bounded bottom fanin over a field \({{\mathbb{F}}}\) of characteristic zero. We resolve this problem by proving a \({N^{\Omega(\frac{d}{\tau})}}\) lower bound for (nonhomogeneous) depth-three arithmetic circuits with bottom fanin at most \({\tau}\) computing an explicit \({N}\)-variate polynomial of degree \({d}\) over \({{\mathbb{F}}}\). Meanwhile, Nisan & Wigderson (Comp Complex 6(3):217–234, 1997) had posed the problem of proving super-polynomial lower bounds for homogeneous depth-five arithmetic circuits. Over fields of characteristic zero, we show a lower bound of \({N^{\Omega(\sqrt{d})}}\) for homogeneous depth-five circuits (resp. also for depth-three circuits) with bottom fanin at most \({N^{\mu}}\), for any fixed \({\mu < 1}\). This resolves the problem posed by Nisan and Wigderson only partially because of the added restriction on the bottom fanin (a general homogeneous depth-five circuit has bottom fanin at most \({N}\)).     

An optimal discrimination of two mixed qubit states with a fixed rate of inconclusive results

In this paper we consider the optimal discrimination of two mixed qubit states for a measurement that allows a fixed rate of inconclusive results. Our strategy is to transform the problem of two qubit states into a minimum error discrimination for three qubit states by adding a specific quantum state \(\rho _{0}\) and a prior probability \(q_{0}\), which behaves as an inconclusive degree. First, we introduce the beginning and the end of practical interval of inconclusive result, \(q_{0}^{(0)}\) and \(q_{0}^{(1)}\), which are key ingredients in investigating our problem. Then we obtain the analytic form of them. Next, we show that our problem can be classified into two cases \(q_{0}=q_{0}^{(0)}\) (or \(q_{0}=q_{0}^{(1)}\)) and \(q_{0}^{(0)}<q_{0}<q_{0}^{(1)}\). In fact, by maximum confidences of two qubit states and non-diagonal element of \(\rho _{0}\), our problem is completely understood. We provide an analytic solution of our problem when \(q_{0}=q_{0}^{(0)}\) (or \(q_{0}=q_{0}^{(1)}\)). However, when \(q_{0}^{(0)}<q_{0}<q_{0}^{(1)}\), we rather supply the numerical method to find the solution, because of the complex relation between inconclusive degree and corresponding failure probability. Finally we confirm our results using previously known examples.     

Enhancing parameter estimation precision by non-Hermitian operator process

Recently, Zhong et al. (Phys Rev A 87:022337, 2013) investigated the dynamics of quantum Fisher information (QFI) in the presence of decoherence. We here reform their results and propose two schemes to enhance and preserve the QFIs for a qubit system subjected to a decoherence noisy environment by applying \({non\text {-}Hermitian}\) operator process either before or after the amplitude damping noise. Resorting to the Bloch sphere representation, we derive the exact analytical expressions of the QFIs with respect to the amplitude parameter \(\theta \) and the phase parameter \(\phi \), and in detail investigate the influence of \({non\text {-}Hermitian}\) operator parameters on the QFIs. Compared with pure decoherence process (without non-Hermitian operator process), we find that the \({post non\text {-}Hermitian}\) operator process can potentially enhance and preserve the QFIs by choosing appropriate \({non\text {-}Hermitian}\) operator parameters, while with the help of the \({prior non\text {-}Hermitian}\) operator process one could completely eliminate the effect of decoherence to improve the parameters estimation. Finally, a generalized non-Hermitian operator parameters effect on the parameters estimation is also discussed.     

Supercloseness of Primal-Dual Galerkin Approximations for Second Order Elliptic Problems

We show that two widely used Galerkin formulations for second-order elliptic problems provide approximations which are actually superclose, that is, their difference converges faster than the corresponding errors. In the framework of linear elasticity, the two formulations correspond to using either the stiffness tensor or its inverse the compliance tensor. We find sufficient conditions, for a wide class of methods (including mixed and discontinuous Galerkin methods), which guarantee a supercloseness result. For example, for the HDG\(_{k}\) method using polynomial approximations of degree \({k>0}\), we find that the difference of approximate fluxes superconverges with order \({k+2}\) and that the difference of the scalar approximations superconverges with order \({k+3}\). We provide numerical results verifying our theoretical results.     

New q-ary quantum MDS codes with distances bigger than $$\frac{ q}{2}$$

The construction of quantum MDS codes has been studied by many authors. We refer to the table in page 1482 of (IEEE Trans Inf Theory 61(3):1474–1484, 2015) for known constructions. However, there have been constructed only a few q-ary quantum MDS \([[n,n-2d+2,d]]_q\) codes with minimum distances \(d>\frac{q}{2}\) for sparse lengths \(n>q+1\). In the case \(n=\frac{q^2-1}{m}\) where \(m|q+1\) or \(m|q-1\) there are complete results. In the case \(n=\frac{q^2-1}{m}\) while \(m|q^2-1\) is neither a factor of \(q-1\) nor \(q+1\), no q-ary quantum MDS code with \(d> \frac{q}{2}\) has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over \(\mathbf{F}_{q^2}\). Then we give some new q-ary quantum codes in this case. Moreover many new q-ary quantum MDS codes with lengths of the form \(\frac{w(q^2-1)}{u}\) and minimum distances \(d > \frac{q}{2}\) are presented.     

Gauge Neural Network with Z(2) Synaptic Variables: Phase Structure and Simulation of Learning and Recalling Patterns

We study the Z(2) gauge-invariant neural network which is defined on a partially connected random network and involves Z(2) neuron variables \(S_i\) (\(=\pm \)1) and Z(2) synaptic connection (gauge) variables \(J_{ij}\) (\(=\pm \)1). Its energy consists of the Hopfield term \(-c_1S_iJ_{ij}S_j\), double Hopfield term \(-c_2 S_iJ_{ij}J_{jk} S_k\), and the reverberation (triple Hopfield) term \(-c_3 J_{ij}J_{jk}J_{ki}\) of synaptic self interactions. For the case \(c_2=0\), its phase diagram in the \(c_3-c_1\) plane has been studied both for the symmetric couplings \(J_{ij}=J_{ji}\) and asymmetric couplings (\(J_{ij}\) and \(J_{ji}\) are independent); it consists of the Higgs, Coulomb and confinement phases, each of which is characterized by the ability of learning and/or recalling patterns. In this paper, we consider the phase diagram for the case of nonvanishing \(c_2\), and examine its effect. We find that the \(c_2\) term enlarges the region of Higgs phase and generates a new second-order transition. We also simulate the dynamical process of learning patterns of \(S_i\) and recalling them and measure the performance directly by overlaps of \(S_i\). We discuss the difference in performance for the cases of Z(2) variables and real variables for synaptic connections.     

Optimal noise functions for location privacy on continuous regions

Users of location-based services are highly vulnerable to privacy risks since they need to disclose, at least partially, their locations to benefit from these services. One possibility to limit these risks is to obfuscate the location of a user by adding random noise drawn from a noise function. In this paper, we require the noise functions to satisfy a generic location privacy notion called \(\ell \)-privacy, which makes the position of the user in a given region \(\mathcal {X}\) relatively indistinguishable from other points in \(\mathcal {X}\). We also aim at minimizing the loss in the service utility due to such obfuscation. While existing optimization frameworks regard the region \(\mathcal {X}\) restrictively as a finite set of points, we consider the more realistic case in which the region is rather continuous with a nonzero area. In this situation, we demonstrate that circular noise functions are enough to satisfy \(\ell \)-privacy on \(\mathcal {X}\) and equivalently on the entire space without any penalty in the utility. Afterward, we describe a large parametric space of noise functions that satisfy \(\ell \)-privacy on \(\mathcal {X}\), and show that this space has always an optimal member, regardless of \(\ell \) and \(\mathcal {X}\). We also investigate the recent notion of \(\epsilon \)-geo-indistinguishability as an instance of \(\ell \)-privacy and prove in this case that with respect to any increasing loss function, the planar Laplace noise function is optimal for any region having a nonzero area.     

Extremal properties of conditional entropy and quantum discord for XXZ,symmetric quantum states

For the XXZ subclass of symmetric two-qubit X states, we study the behavior of quantum conditional entropy \(S_{cond}\) as a function of measurement angle \(\theta \in [0,\pi /2]\). Numerical calculations show that the function \(S_{cond}(\theta )\) for X states can have at most one local extremum in the open interval from zero to \(\pi /2\) (unimodality property). If the extremum is a minimum, the quantum discord displays region with variable (state-dependent) optimal measurement angle \(\theta ^*\). Such \(\theta \)-regions (phases, fractions) are very tiny in the space of X-state parameters. We also discover the cases when the conditional entropy has a local maximum inside the interval \((0,\pi /2)\). It is remarkable that the maxima exist in surprisingly wide regions, and the boundaries for such regions are defined by the same bifurcation conditions as for those with a minimum.     

The remote set problem on lattices

We initiate studying the Remote Set Problem (\({\mathsf{RSP}}\)) on lattices, which given a lattice asks to find a set of points containing a point which is far from the lattice. We show a polynomial-time deterministic algorithm that on rank n lattice \({\mathcal{L}}\) outputs a set of points, at least one of which is \({\sqrt{\log n / n} \cdot \rho(\mathcal{L})}\) -far from \({\mathcal{L}}\) , where \({\rho(\mathcal{L})}\) stands for the covering radius of \({\mathcal{L}}\) (i.e., the maximum possible distance of a point in space from \({\mathcal{L}}\)). As an application, we show that the covering radius problem with approximation factor \({\sqrt{n / \log n}}\) lies in the complexity class \({\mathsf{NP}}\) , improving a result of Guruswami et al. (Comput Complex 14(2): 90–121, 2005) by a factor of \({\sqrt{\log n}}\) .Our results apply to any \({\ell_p}\) norm for \({2 \leq p \leq \infty}\) with the same approximation factors (except a loss of \({\sqrt{\log \log n}}\) for \({p = \infty}\)). In addition, we show that the output of our algorithm for \({\mathsf{RSP}}\) contains a point whose \({\ell_2}\) distance from \({\mathcal{L}}\) is at least \({(\log n/n)^{1/p} \cdot \rho^{(p)}(\mathcal{L})}\) , where \({\rho^{(p)}(\mathcal{L})}\) is the covering radius of \({\mathcal{L}}\) measured with respect to the \({\ell_p}\) norm. The proof technique involves a theorem on balancing vectors due to Banaszczyk (Random Struct Algorithms 12(4):351–360, 1998) and the “six standard deviations” theorem of Spencer (Trans Am Math Soc 289(2):679–706, 1985).     

A general framework for architecture composability

Architectures depict design principles: paradigms that can be understood by all, allow thinking on a higher plane and avoiding low-level mistakes. They provide means for ensuring correctness by construction by enforcing global properties characterizing the coordination between components. An architecture can be considered as an operator A that, applied to a set of components \({\mathcal{B}}\), builds a composite component \({A(\mathcal{B})}\) meeting a characteristic property \({\Phi}\). Architecture composability is a basic and common problem faced by system designers. In this paper, we propose a formal and general framework for architecture composability based on an associative, commutative and idempotent architecture composition operator \({\oplus}\). The main result is that if two architectures A ₁ and A ₂ enforce respectively safety properties \({\Phi_{1}}\) and \({\Phi_{2}}\), the architecture \({A_{1} \oplus A_{2}}\) enforces the property \({\Phi_{1} \land \Phi_{2}}\), that is both properties are preserved by architecture composition. We also establish preservation of liveness properties by architecture composition. The presented results are illustrated by a running example and a case study.     

Data assimilation and sampling in Banach spaces

This paper studies the problem of approximating a function f in a Banach space \(\mathcal{X}\) from measurements \(l_j(f)\), \(j=1,\ldots ,m\), where the \(l_j\) are linear functionals from \(\mathcal{X}^*\). Quantitative results for such recovery problems require additional information about the sought after function f. These additional assumptions take the form of assuming that f is in a certain model class \(K\subset \mathcal{X}\). Since there are generally infinitely many functions in K which share these same measurements, the best approximation is the center of the smallest ball B, called the Chebyshev ball, which contains the set \(\bar{K}\) of all f in K with these measurements. Therefore, the problem is reduced to analytically or numerically approximating this Chebyshev ball. Most results study this problem for classical Banach spaces \(\mathcal{X}\) such as the \(L_p\) spaces, \(1\le p\le \infty \), and for K the unit ball of a smoothness space in \(\mathcal{X}\). Our interest in this paper is in the model classes \(K=\mathcal{K}(\varepsilon ,V)\), with \(\varepsilon >0\) and V a finite dimensional subspace of \(\mathcal{X}\), which consists of all \(f\in \mathcal{X}\) such that \(\mathrm{dist}(f,V)_\mathcal{X}\le \varepsilon \). These model classes, called approximation sets, arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance and algorithms for finding near optimal approximations. It builds on the initial analysis given in Maday et al. (Int J Numer Method Eng 102:933–965, 2015) for the case when \(\mathcal{X}\) is a Hilbert space, and further studied in Binev et al. (SIAM UQ, 2015). It is shown how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation.     

Hybridized Discontinuous Galerkin Method for Elliptic Interface Problems: Error Estimates Under Low Regularity Assumptions of Solutions

New hybridized discontinuous Galerkin (HDG) methods for the interface problem for elliptic equations are proposed. Unknown functions of our schemes are \(u_h\) in elements and \(\hat{u}_h\) on inter-element edges. That is, we formulate our schemes without introducing the flux variable. We assume that subdomains \(\Omega _1\) and \(\Omega _2\) are polyhedral domains and that the interface \(\Gamma =\partial \Omega _1\cap \partial \Omega _2\) is polyhedral surface or polygon. Moreover, \(\Gamma \) is assumed to be expressed as the union of edges of some elements. We deal with the case where the interface is transversely connected with the boundary of the whole domain \(\overline{\Omega }=\overline{\Omega _1\cap \Omega _2}\). Consequently, the solution u of the interface problem may not have a sufficient regularity, say \(u\in H^2(\Omega )\) or \(u|_{\Omega _1}\in H^2(\Omega _1)\), \(u|_{\Omega _2}\in H^2(\Omega _2)\). We succeed in deriving optimal order error estimates in an HDG norm and the \(L^2\) norm under low regularity assumptions of solutions, say \(u|_{\Omega _1}\in H^{1+s}(\Omega _1)\) and \(u|_{\Omega _2}\in H^{1+s}(\Omega _2)\) for some \(s\in (1/2,1]\), where \(H^{1+s}\) denotes the fractional order Sobolev space. Numerical examples to validate our results are also presented.     

Mutually unbiased maximally entangled bases in $$\mathbb {C}^d\otimes \mathbb {C}^d$$

We study mutually unbiased maximally entangled bases (MUMEB’s) in bipartite system \(\mathbb {C}^d\otimes \mathbb {C}^d (d \ge 3)\). We generalize the method to construct MUMEB’s given in Tao et al. (Quantum Inf Process 14:2291–2300, 2015), by using any commutative ring R with d elements and generic character of \((R,+)\) instead of \(\mathbb {Z}_d=\mathbb {Z}/d\mathbb {Z}\). Particularly, if \(d=p_1^{a_1}p_2^{a_2}\ldots p_s^{a_s}\) where \(p_1, \ldots , p_s\) are distinct primes and \(3\le p_1^{a_1}\le \cdots \le p_s^{a_s}\), we present \(p_1^{a_1}-1\) MUMEB’s in \(\mathbb {C}^d\otimes \mathbb {C}^d\) by taking \(R=\mathbb {F}_{p_1^{a_1}}\oplus \cdots \oplus \mathbb {F}_{p_s^{a_s}}\), direct sum of finite fields (Theorem 3.3).     

A note on double lacunary statistical $$\sigma$$-convergence of fuzzy numbers

Quite recently, Sava? (Appl Math Lett 21:134–141, 2008), defined the lacunary statistical analogue for double sequence \(X=\{X_{k,l}\}\) of fuzzy numbers as follows: a double sequence \(X=\{X_{k,l}\}\) is said to be lacunary P-statistically convergent to \(X_{0}\) provided that for each \(\epsilon >0\)

$ P-\lim_{r,s}\frac{1}{h_{r,s}}\left | \{(k,l)\in I_{r,s}: d(X_{k,l },X_0)\geq \epsilon\}\right|= 0. $

In this paper we introduce and study double lacunary \(\sigma\)-statistical convergence for sequence of fuzzy numbers and also we get some inclusion theorems.

     

On topological systems

We establish the reflectivity of the subcategories of \(T_{0}\) and sober topological systems in the category \(\mathbf {TopSys}\) of topological systems. We also introduce a Sierpinski object in the category \(\mathbf {TopSys}\) and point out its connection with \(T_{0}\) and sober topological systems and also with injective \(T_{0}\)-topological systems.     

Subspace Learning by $$\ell ^{0}$$-Induced Sparsity

Subspace clustering methods partition the data that lie in or close to a union of subspaces in accordance with the subspace structure. Such methods with sparsity prior, such as sparse subspace clustering (SSC) (Elhamifar and Vidal in IEEE Trans Pattern Anal Mach Intell 35(11):2765–2781, 2013) with the sparsity induced by the \(\ell ^{1}\)-norm, are demonstrated to be effective in subspace clustering. Most of those methods require certain assumptions, e.g. independence or disjointness, on the subspaces. However, these assumptions are not guaranteed to hold in practice and they limit the application of existing sparse subspace clustering methods. In this paper, we propose \(\ell ^{0}\)-induced sparse subspace clustering (\(\ell ^{0}\)-SSC). In contrast to the required assumptions, such as independence or disjointness, on subspaces for most existing sparse subspace clustering methods, we prove that \(\ell ^{0}\)-SSC guarantees the subspace-sparse representation, a key element in subspace clustering, for arbitrary distinct underlying subspaces almost surely under the mild i.i.d. assumption on the data generation. We also present the “no free lunch” theorem which shows that obtaining the subspace representation under our general assumptions can not be much computationally cheaper than solving the corresponding \(\ell ^{0}\) sparse representation problem of \(\ell ^{0}\)-SSC. A novel approximate algorithm named Approximate \(\ell ^{0}\)-SSC (A\(\ell ^{0}\)-SSC) is developed which employs proximal gradient descent to obtain a sub-optimal solution to the optimization problem of \(\ell ^{0}\)-SSC with theoretical guarantee. The sub-optimal solution is used to build a sparse similarity matrix upon which spectral clustering is performed for the final clustering results. Extensive experimental results on various data sets demonstrate the superiority of A\(\ell ^{0}\)-SSC compared to other competing clustering methods. Furthermore, we extend \(\ell ^{0}\)-SSC to semi-supervised learning by performing label propagation on the sparse similarity matrix learnt by A\(\ell ^{0}\)-SSC and demonstrate the effectiveness of the resultant semi-supervised learning method termed \(\ell ^{0}\)-sparse subspace label propagation (\(\ell ^{0}\)-SSLP).