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Guaranteed Matrix Completion via Non-convex Factorization

Ruoyu Sun, Zhi-Quan Luo

arXiv:1411.8003v3cs.LG

TL;DR

The paper addresses limited theoretical understanding of efficient but non-convex matrix-factorization methods for matrix completion. It studies the local geometry of a properly regularized formulation and proves that many standard algorithms converge to global optima and exactly recover the matrix without resampling. The analysis still leaves open whether incoherence is necessary for the local geometry when the sample size is large.

  • Problem

    Whether standard algorithms for non-convex matrix-factorization formulations can recover the true low-rank matrix remains largely open despite their empirical success.

  • Method

    The paper analyzes the local geometry of a regularized factorization objective and combines this with an invariant-region convergence argument for standard optimization algorithms.

  • Results

    Many standard algorithms converge to global optima and reconstruct M exactly with high probability when sufficiently many entries are revealed, without resampling.

  • Takeaways & Limitations

    The guarantees extend exact-recovery theory to gradient descent, SGD, and alternating-minimization-type methods using standard algorithm forms without resampling.

  • Takeaways & Limitations

    When the number of samples is large, it remains open whether incoherence is required for the local geometry or whether algorithms automatically control row norms through an unknown mechanism.

Abstract

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Matrix factorization is a popular approach for large-scale matrix completion. The optimization formulation based on matrix factorization can be solved very efficiently by standard algorithms in practice. However, due to the non-convexity caused by the factorization model, there is a limited theoretical understanding of this formulation. In this paper, we establish a theoretical guarantee for the factorization formulation to correctly recover the underlying low-rank matrix. In particular, we show that under similar conditions to those in previous works, many standard optimization algorithms converge to the global optima of a factorization formulation, and recover the true low-rank matrix. We study the local geometry of a properly regularized factorization formulation and prove that any stationary point in a certain local region is globally optimal. A major difference of our work from the existing results is that we do not need resampling in either the algorithm or its analysis. Compared to other works on nonconvex optimization, one extra difficulty lies in analyzing nonconvex constrained optimization when the constraint (or the corresponding regularizer) is not "consistent" with the gradient direction. One technical contribution is the perturbation analysis for non-symmetric matrix factorization.

1 Introduction

Matrix completion uses low-rank structure through nuclear-norm or matrix-factorization formulations, but factorization methods lack sufficient theory despite practical efficiency. This paper analyzes their local geometry and proves recovery guarantees for many standard algorithms without resampling.

  • Problem setting: Matrix completion recovers an unknown low-rank matrix from a subset of observed entries, exploiting low-rank structure for large-scale data.The nuclear-norm formulation has exact-recovery guarantees under incoherence with O(r(m + n) log^2(m + n)) revealed entries.
  • Factorization approach: Matrix factorization represents the unknown rank-r matrix as XY^T with smaller factors, automatically enforcing the low-rank requirement.Its regularized square-loss formulation is non-convex and can be optimized to stationary points using gradient descent, alternating minimization, SGD, and related methods.
  • Paper contributions: The paper addresses the open question of whether standard factorization algorithms can recover the true low-rank matrix under conditions similar to prior work.The analyzed class includes gradient descent, SGD, two-block alternating minimization, and block coordinate gradient descent.
  • Paper contributions: The analysis establishes local geometry for a regularized factorization objective, aligning local gradient directions with global descent directions while handling incoherence constraints and non-symmetric factors.A second regularizer corrects the descent-direction issue created by the incoherence regularizer.
  • Implications: The results apply to standard algorithm forms without independent resampling across iterations, and the sample-complexity bound is independent of the recovery error ϵ.Experiments associate balanced factor rows with successful instances and report that adding G2 can bring sample complexity close to the fundamental limit on synthetic Gaussian data.
  • Proof strategy: The proof combines a no-spurious-stationary-point property in a local region with an algorithmic guarantee that standard iterates remain in that region and converge to stationary points.Global optimality in the relevant region implies exact recovery, equivalently that zero training error yields zero generalization error.

2 Problem Formulation and Algorithms

The paper formulates matrix completion with a regularized factorization model, assumes incoherence and random sampling, and analyzes initialization and several optimization algorithms. The hard regularizer preserves the optimal solution while controlling factor incoherence and norms.

  • 2.1 Assumptions: The matrix is assumed to be µ-incoherent, meaning its compact-SVD factors have bounded row norms.The paper uses this standard assumption for low-rank recovery and also applies “incoherent” to factor matrices with bounded row norms.
  • 2.1 Assumptions: The observed index set Ω is uniformly random with fixed size, and the analysis equivalently uses independent Bernoulli sampling.The two sampling models are treated as equivalent for high-probability success statements under the corresponding sampling rate.
  • 2.2 Problem formulation: The formulation adds smooth penalty terms that control factor row norms and overall factor norms, enforcing incoherence and boundedness.The regularization function is built from a convex, continuously differentiable scalar penalty that vanishes within the feasible range.
  • 2.2 Problem formulation: The hard regularizer is a penalty formulation for constrained optimization and does not distort the optimal solution.Its convexity follows from composing non-decreasing convex functions with convex norm-based terms.
  • 2.2 Problem formulation: The true factors achieve objective value zero, so the regularized problem retains an optimal factorization of the incoherent target matrix.The square-loss term vanishes when XY^T=M, and the regularizer also vanishes for the corresponding SVD factors.
  • 2.2 Problem formulation: Within a bounded region, the regularized objective has Lipschitz-continuous gradients, supporting standard gradient-based analysis.The paper states this property with a Lipschitz constant L(β0) over Γ(β0).
  • 2.3 Row-scaled Spectral Initialization: Initialization uses a spectral method followed by row scaling to produce a point in the required incoherent neighborhood.The paper states that the initial point must be sufficiently close to the global optima and presents this procedure in Table 1.
  • 2.4 Algorithms: The guarantee covers gradient descent, SGD, and block-coordinate methods, including alternating minimization and BCGD; regularization can activate mainly in sparse-observation regimes.For alternating minimization, the regularized subproblem may require an inner loop, while one gradient step yields BCGD, which retains exact recovery guarantees.

3 Main Results

Under random sampling and suitable initialization, the regularized factorization formulation has local geometry that drives standard algorithms toward global optima and exact recovery. The guarantees include high-probability recovery, linear convergence for gradient descent, and an explicit sample-complexity boundary.

  • Exact recovery: Algorithms 1–4 converge to global optima and reconstruct M exactly with high probability when sufficiently many entries are revealed.The guarantee applies under the conditions of Theorem 3.1 and random sampling of Ω.
  • Algorithmic scope: The result covers gradient descent, SGD, alternating minimization, and block coordinate methods without requiring independent samples across iterations.This provides a recovery guarantee for standard factorization algorithms without resampling.
  • Sample complexity: For constant rank and bounded condition number, the sample complexity is order optimal at O(n log n), while for r = O(log n) it is suboptimal by a polylogarithmic factor.The initialization contributes an r^4κ^4 factor, and the authors expect alternative initialization procedures could reduce these exponents.
  • Convergence speed: Gradient descent converges linearly, and the same rate extends to algorithms satisfying the sufficient-decrease condition together with the required local-region properties.The proof uses a cost-to-go estimate and sufficient descent; alternating-type methods can satisfy sufficient decrease under mild conditions.
  • Convergence speed: The resulting iteration complexity is ˜O(poly(n) log 1/ϵ), while improving this bound to ˜O(|Ω| log(1/ϵ)) is left for future work.The hidden polynomial factors depend on r, κ, and α.
  • Local geometry: Any stationary point in K1 ∩ K2 ∩ K(δ) satisfies XY^T = M, establishing the local geometric property underlying exact recovery.The iterates remain in this local region, allowing the stationary-point result to be applied to algorithmic limit points.

4 Proof of Lemma 3.1

The proof constructs an incoherent factorization M = UV^T near XY^T and verifies three inequalities that establish the local geometric property needed for Lemma 3.1.

  • Proof strategy: The proof reduces Lemma 3.1 to constructing U, V so two sampling-error norms are bounded and the regularizer-gradient inner product is nonnegative.These are the three inequalities used to establish the desired local geometry.
  • Proof strategy: The decomposition XY^T − M = a + b separates a component controlled by fixed row or column spaces from a component controlled using a random graph lemma.This handles the dependence between the error matrix and the sampling set Ω.
  • Construction of U, V: Proposition 4.1 gives a factorization M = UV^T close to X, Y with incoherent factors when XY^T is close to M.It establishes continuity of the factorization map near the low-rank matrix M.
  • Construction of U, V: Proposition 4.2 strengthens the construction when X and Y have large norms by additionally bounding the norms of U and V relative to the original factors.The stronger norm contraction is used for linear-convergence analysis.
  • Key inequalities: The constructed factors satisfy the required inequalities: (40b) by Corollary 4.2, (40a) by Proposition 4.3, and (37b) by Claim 4.1.Claim 4.1 establishes φ_G = ⟨∇_XG, X−U⟩ + ⟨∇_YG, Y−V⟩ ≥ 0.

5 Proof of Lemma 3.2

The proof shows that Algorithms 1–4 converge to stationary points while remaining in the required incoherent local region, thereby establishing Lemma 3.2.

  • Stationarity: Property (a) requires limit points of the algorithmic sequences to be stationary points, and Claim 5.1 proves this for Algorithms 1–4.This part uses classical optimization results.
  • Initialization: The initialization lies in a scaled incoherent neighborhood where the regularizer vanishes, as established by Claim 5.2.The scaling ensures G(X_0, Y_0) = 0.
  • Algorithm conditions: Algorithm 1 satisfies the line-search or stepsize condition, while Algorithms 2 and 3 use exact convex upper-bound minimization and Algorithm 4 satisfies the bounded-step condition.Claim 5.3 maps each algorithm to one of the conditions in (67).
  • Conclusion: Combining stationarity with local-region invariance proves that Algorithms 1–4 satisfy both properties required by Lemma 3.2.The argument combines Claims 5.1–5.3 with Proposition 5.1.

A.1 Proof of Claim 2.1

The appendix proves Claim 2.1 through norm inequalities, derivative bounds, and row-wise subproblem analysis for the regularized objective and its algorithmic updates.

  • Norm bounds: The proof bounds objective terms using triangle inequalities and perturbation norms involving X−U, Y−V, and the sampled residual.The bounds exploit assumptions on factor norms and the size of M.
  • Regularizer bounds: The regularizer analysis controls first and second derivatives through row-wise structure and bounded factor norms.The argument applies mean-value and triangle inequalities to the regularizer terms.
  • Conclusion: Combining the derivative and norm estimates yields the claimed bound in Claim 2.1.The proof concludes after substituting the intermediate inequalities.
  • Refinement: With row norms additionally bounded by O(β1) and O(β2), the Lipschitz constant can be improved to 4β2.This is stated as a stronger conditional estimate.
  • Row-wise updates: Algorithm 3 solves each row subproblem using an increasing scalar equation and a simple bisection procedure, with an analogous procedure for Y rows.Positive semidefiniteness makes the relevant matrix properties suitable for this solution method.

B.2 Proof of Proposition 4.1

The proof of Proposition 4.1 constructs U and V from the SVD of M so that UV^T = M while preserving closeness, norm control, and incoherence-related properties.

  • Construction: The construction starts from the SVD of M and decomposes X and Y into components aligned with and orthogonal to M’s singular spaces.The aligned components are used to build factors representing M exactly.
  • Construction: Invertibility of the aligned factor blocks follows from the closeness of XY^T to M and lower bounds involving Σ_min.These bounds ensure the relevant rank-r components remain nonsingular.
  • Verified properties: The resulting factors satisfy UV^T = M and the required norm contraction for U relative to X.The proof verifies requirement (48a) and (48b) directly from the construction.
  • Perturbation bounds: The proof bounds the factor perturbations using the decomposed aligned and orthogonal components, with factor norms controlled by β_T.These estimates establish the closeness requirements in Proposition 4.1.
  • Incoherence: The incoherence assumption on M controls the row norms of the constructed factors through the SVD-based representation.The argument uses norm inequalities for products involving incoherent singular-vector matrices.

C Proof of Proposition 4.2

The proof reduces Proposition 4.2 to an r × r matrix proposition, then transfers its factorization and norm conclusions back to the original matrices.

  • Reduction to Proposition C.1: Proposition C.1 is the central reduced statement used to prove Proposition 4.2.The reduction to r × r matrices is described as straightforward, while Proposition C.1 contains the major difficulty.
  • Change of variables: The proof represents the original matrices through SVD factors and transformed r × r variables.Unitary SVD factors Q1 and Q2 are used to express X and Y and convert conditions on U,V into conditions on reduced variables.
  • Reduced proposition: Proposition C.1 supplies reduced factors U′, V′ satisfying the required relations and bounds under its stated assumptions.The proposition introduces numerical constants and assumes conditions involving X′, Y′, and βT.
  • Lifting the conclusion: The conclusion transfers back by defining U,V from the reduced factors and verifying UVT = M.Requirement (50a) follows directly from (114a) and the definitions of U and V.
  • Lifting the conclusion: The remaining lifted conditions are established through Frobenius- and spectral-norm inequalities for U,V and their differences from X,Y.The proof separately checks requirements (50b), (50c), and (50d), including an analogous bound for ∥V − Y∥F.

C.2.1 Perturbation Analysis for Preconditioning.

This section interprets the perturbation proposition as a preconditioning result and develops two motivating constructions: shrinking/extending imbalanced entries and rotating vectors with balanced row norms.

  • Preconditioning interpretation: Proposition C.1 is related to preconditioning because it constructs nearby factors with improved conditioning.Preconditioning is described as reducing the condition number through preprocessing.
  • Preconditioning interpretation: Proposition C.2 shows that for non-singular X with ∥X∥F = ∥X^-1∥F ≥ C√r, a small perturbation can reduce the inverse norm while remaining close to X.The perturbation distance is O(d′r^1.5), with d′ ≤ O(1/r^1.5).
  • Preconditioning interpretation: Under d ≤ O(Σmin/r) and equal factor norms β, Proposition C.3 yields a factorization Σ = UVT close to X,Y with no larger factor norms.The distance bound is O(√r d β/Σmin).
  • First motivating example: In the first example, shrinking large diagonal entries and extending small ones preserves the Frobenius norm while increasing the relevant products.The construction produces more balanced diagonal factors U,V.
  • Second motivating example: The first operation fails when row lengths are balanced, so the second example uses equal-norm rotations to modify inner products without violating norm constraints.Rotating y1 and x2 by the same angle preserves norms and maintains the required orthogonality structure.

C.2.3 Proof Ideas of Proposition C.1

The proof constructs an intermediate factor satisfying the product constraint first, then modifies the factors to meet norm constraints using either shrinking/extending or rotation.

  • Proof framework: The two motivating operations are individually suited to different angle and row-imbalance regimes.Shrinking/extending handles sufficiently small angles, while rotation handles sufficiently large angles.
  • Proof framework: The proof avoids applying both operations simultaneously because that approach produces bounds with a large exponent of r.The alternative framework uses only one operation after constructing an intermediate factor.
  • Two-step construction: It first defines Ȳ so that XȲT = Σ, satisfying the product requirement before addressing factor norms.This isolates the exact product constraint from the subsequent norm-reduction step.
  • Case split: The case split depends on whether most angles between X and Ȳ are below or above a threshold, selecting shrinking/extending or rotation accordingly.This replaces the earlier need to modify every row pair simultaneously with an overall norm-reduction argument.
  • Two-step construction: If Ȳ has excessive norm, the proof modifies X and Ȳ to U,V while preserving their product and reducing both factor norms to the required levels.The construction targets ∥V∥F ≤ ∥Y∥F and ∥U∥F ≤ ∥X∥F.

C.3.1 Proof of Case 1

Case 1 constructs U,V by shrinking and extending selected row vectors of X and the intermediate factor Ȳ, preserving the product while enforcing norm bounds.

  • Construction: The construction targets Case 1 by modifying X and Ȳ through a shrinking/extending operation.The precise definition of U and V is given in Table 8.
  • Product preservation: The operation preserves the product constraint because XȲT = Σ before modification and the row-wise construction maintains this relation.Requirement (114a) follows directly from the definition of U,V and XȲT = Σ.
  • Norm and distance bounds: The proof verifies the required norm and distance conditions for U,V using bounds on the affected row norms and perturbations.Requirements (114b) and (114c) are established through inequalities involving η, η̄, and the row norms.
  • Case analysis: The case analysis bounds aggregate quantities for selected row groups and uses these bounds to establish the key inequalities.The proof derives relations such as S2 ≤ S1/2 and S3 ≤ 3S1/4 before proving the norm requirement.
  • Case analysis: A contradiction argument establishes the required threshold relations, completing the verification of the first part of (114b).The contradiction uses the definition CT = 20 and derives an incompatible bound CT < 389.

C.3.2 Proof of Case 2a

Case 2a constructs a rotation-and-shrinking operation that reduces Y's norm while preserving the factorization and relevant inner products. The proof establishes that the operation exists and maintains the required angle, norm, and orthogonality properties.

  • C.3.2 Proof of Case 2a: For sufficiently small Di, Operation 2 produces Xi,Yi satisfying the properties summarized in Claim C.2.The claim provides the recursive invariants needed for the subsequent factorization argument.
  • C.3.2 Proof of Case 2a: The recursively constructed factors preserve ∥Xi∥F = ∥X∥F while shrinking the Y norm according to the prescribed scaling relations.The proof uses properties (147b) and (147d)–(147f) to establish the requirements on U and V.
  • C.3.2 Proof of Case 2a: Operation 2 rotates and shrinks selected rows of X and Y while preserving ⟨X,Y⟩ and ∥X∥F.Rows not selected for updating remain unchanged, and the construction preserves orthogonality between the factors.
  • C.3.2 Proof of Case 2a: The construction ensures that rotating yj increases ⟨xj,yj⟩ for every j ≠ i, relying on ⟨xi,yi⟩ > 0.The geometric argument explains why the required rotations preserve the desired inner-product behavior.
  • C.3.2 Proof of Case 2a: Each subsequent angle decreases by at most 1/(24sπ), allowing Operation 2 to be applied repeatedly while retaining a positive angular margin.The proof bounds the decrease in later angles by the decrease in the updated angle.

D.1 Proof of Claim 5.2

The proof of Claim 5.2 establishes that the SVD-based initialization lies in the required incoherent and bounded neighborhoods. It combines approximation of the observed matrix with scaling and perturbation bounds.

  • D.1 Proof of Claim 5.2: The proof first controls the distance between M and its best rank-r approximation M0, then shows scaling preserves the needed closeness.This two-part strategy connects spectral approximation of the observed matrix to the factorized initialization.
  • D.1 Proof of Claim 5.2: Lemma D.1 supplies a high-probability approximation bound for M0 under the stated random sampling condition.The lemma assumes entrywise sampling probability p and rank-r structure, with constants governing the approximation guarantee.
  • D.1 Proof of Claim 5.2: The initialization satisfies the first neighborhood property directly from the definitions of X0 and Y0.The argument identifies this property with membership in the set associated with the initialization bounds.
  • D.1 Proof of Claim 5.2: The approximation bound ∥M−M0∥F ≤ Σmin/5 ≤ Σmax/5 establishes the second neighborhood property for sufficiently large C0.The proof uses the spectral norm lower bound and the definition of X0 to control its factor norms.
  • D.1 Proof of Claim 5.2: Proposition 4.1 then provides nearby factors U0,V0 whose deviations from X0,Y0 are bounded.These perturbation bounds complete the initialization argument.

D.3 Proof of Proposition 5.1

The proof of Proposition 5.1 shows that low objective value forces iterates into the incoherent and bounded region and keeps them close to a global optimum. This supports the local optimization analysis.

  • D.3 Proof of Proposition 5.1: A general function-value condition is used to establish membership in K1 ∩ K2, the region of incoherent and bounded factors.The argument derives this implication through Proposition D.1 under the stated sampling assumptions.
  • D.3 Proof of Proposition 5.1: If a point violates K1 ∩ K2, the regularizer makes its objective exceed the allowed objective level, yielding a contradiction.The proof lower-bounds the regularization contribution when row or Frobenius norm constraints fail.
  • D.3 Proof of Proposition 5.1: The objective along the considered updates remains nonincreasing and bounded by twice its initial value.This bound is used to invoke Proposition D.1 at every iteration.
  • D.3 Proof of Proposition 5.1: Lemma D.2 shows that any point with objective at most 2F̃(x0) lies within the prescribed distance of a global optimum.The contradiction argument forces equality of the objective values and zero distance in the limiting case.
  • D.3 Proof of Proposition 5.1: An induction over iterations proves that all iterates remain inside the target neighborhood K(2δ/3).The proof treats the three update cases separately and derives a contradiction whenever an iterate leaves the neighborhood.

D.4 Proof of Claim 5.3

The proof of Claim 5.3 verifies that the optimization algorithms satisfy the descent and boundedness conditions required for the local convergence analysis. It handles deterministic updates and SGD through Lipschitz and gradient-error bounds.

  • D.4 Proof of Claim 5.3: Algorithms 2 and 3 satisfy the required update condition because they exactly minimize convex upper bounds along selected directions.Their iteration indexing is aligned with the analysis by treating one block update as an iteration.
  • D.4 Proof of Claim 5.3: Algorithm 1 with a sufficiently small constant stepsize satisfies the descent-path condition by the descent lemma and Lipschitz continuity.The proof maintains bounded iterates so the gradient remains Lipschitz along each update segment.
  • D.4 Proof of Claim 5.3: The iterate norms remain bounded through an induction over component updates, placing each new iterate in a common bounded set.Uniform gradient bounds control the norm increase at each component update.
  • D.4 Proof of Claim 5.3: SGD is analyzed as approximate gradient descent, with the error controlled by differences between component gradients at intermediate and current iterates.The component gradients are Lipschitz, allowing the accumulated approximation error to be bounded.
  • D.4 Proof of Claim 5.3: With an appropriate stepsize, SGD satisfies the same descent-path condition required by the convergence framework.The proof concludes this after combining the boundedness argument with the descent lemma.

D.5 Proof of Claim 5.1

The proof establishes that several algorithms generate limit points that are stationary for the regularized factorization problem, including constant-step, line-search, block-coordinate, BSUM, and SGD methods.

  • Algorithm 1: Constant-step Algorithm 1 remains in a bounded region because its objective value decreases and the relevant gradient is Lipschitz continuous there.The iterates lie in K1 ∩ K2, where the gradient has Lipschitz constant L(βT).
  • Algorithm 1: The restricted Armijo analysis proves that its initial stepsizes are asymptotically lower bounded, enabling the standard convergence argument.The proof establishes lower boundedness by contradiction: vanishing initial stepsizes would force unbounded gradients in a region where gradients are bounded.
  • Algorithm 1: The restricted line-search rule inherits the convergence properties of the restricted Armijo rule because it achieves at least as much cost reduction per iteration.Thus the proof transfers stationarity of limit points from the Armijo variant to the restricted line-search variant.
  • Algorithm 2: Each limit point of the two-block BCD Algorithm 2 is a stationary point of problem (P1).This follows from the cited convergence result for the two-block BCD method.
  • Algorithm 3: Each limit point of BSUM Algorithm 3 is stationary because its compact level set and strongly convex subproblems satisfy the required BSUM conditions.Each subproblem uses a convex tight upper bound and therefore has a unique solution.
  • Algorithm 4: Each limit point of the SGD-style Algorithm 4 is stationary because the method can be viewed as approximate gradient descent with bounded error.The conclusion uses the stated stepsize conditions and a cited result for bounded-error gradient methods.

E Proof of Lemma 3.3

The proof of Lemma 3.3 establishes high-probability bounds for the observed-data component and deterministic bounds for the fully observed component over the specified local region.

  • Lemma statement: With probability at least 1 − 1/n^4, the lemma's stronger statement holds uniformly for (X, Y) in K1 ∩ K2 ∩ K(δ).The statement applies to U and V defined in Table 7.
  • Notation: The proof introduces d = ∥M − XY^T∥F as the factorization error used in subsequent bounds.The quantity d appears in the case analysis and algebraic estimates.
  • Probability structure: The bound on φF holds with high probability because φF depends on the observed set Ω, whereas the bound on φG always holds because φG is independent of Ω.This separates the sampling-dependent and sampling-independent parts of the argument.
  • Claim E.1: Claim E.1 bounds the component contributions of G for X and Y over K1 ∩ K2 ∩ K(δ), and its proof proceeds by analyzing two cases.The component functions are introduced through the decomposition of G(X, Y).
  • Case analysis: The proof handles the X-side inequality through separate norm cases, including zero-gradient cases and cases controlled by Corollary 4.1.The Y-side inequality follows by a symmetric argument when the ordering of ∥X∥F and ∥Y∥F is reversed.
  • Case analysis: The remaining cases use bounds on U and V involving d/Σmin, after which the inequalities are combined algebraically to finish Lemma 3.3.The proof explicitly concludes the case inequality and then completes the lemma through substitutions and simplification.
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